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2549 lines
109 KiB
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<a href="../MathNet.Numerics/AppSwitches.htm">AppSwitches</a>
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<a href="../MathNet.Numerics/Combinatorics.htm">Combinatorics</a>
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<a href="../MathNet.Numerics/ExcelFunctions.htm">ExcelFunctions</a>
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<a href="../MathNet.Numerics/FindMinimum.htm">FindMinimum</a>
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<a href="../MathNet.Numerics/FindRoots.htm">FindRoots</a>
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<a href="../MathNet.Numerics/IPrecisionSupport`1.htm">IPrecisionSupport<T></a>
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<a href="../MathNet.Numerics/MemoryAllocationException.htm">MemoryAllocationException</a>
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<a href="../MathNet.Numerics/NonConvergenceException.htm">NonConvergenceException</a>
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<a href="../MathNet.Numerics/NumericalBreakdownException.htm">NumericalBreakdownException</a>
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<a href="../MathNet.Numerics/Permutation.htm">Permutation</a>
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<a href="../MathNet.Numerics/Polynomial.htm">Polynomial</a>
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<a href="../MathNet.Numerics/Precision.htm">Precision</a>
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<a href="../MathNet.Numerics/Series.htm">Series</a>
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<a href="../MathNet.Numerics/SingularUMatrixException.htm">SingularUMatrixException</a>
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<a href="../MathNet.Numerics/Sorting.htm">Sorting</a>
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<a href="../MathNet.Numerics/SpecialFunctions.htm" class="current">SpecialFunctions</a>
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<li>
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<a href="../MathNet.Numerics/TestFunctions.htm">TestFunctions</a>
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</li>
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<li>
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<a href="../MathNet.Numerics/Trig.htm">Trig</a>
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</li>
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<a href="../MathNet.Numerics/Window.htm">Window</a>
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</ul>
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</div>
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</div>
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<div class="header">
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<p class="class"><strong>Type</strong> SpecialFunctions</p>
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<p><strong>Namespace</strong> MathNet.Numerics</p>
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</div>
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<div class="sub-header">
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<div id="summary">This partial implementation of the SpecialFunctions class contains all methods related to the Airy functions.
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</div>
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<h3 class="section">Static Functions</h3>
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<ul>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#AiryAi">AiryAi</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#AiryAi">AiryAi</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#AiryAiPrime">AiryAiPrime</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#AiryAiPrime">AiryAiPrime</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#AiryAiPrimeScaled">AiryAiPrimeScaled</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#AiryAiPrimeScaled">AiryAiPrimeScaled</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#AiryAiScaled">AiryAiScaled</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#AiryAiScaled">AiryAiScaled</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#AiryBi">AiryBi</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#AiryBi">AiryBi</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#AiryBiPrime">AiryBiPrime</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#AiryBiPrime">AiryBiPrime</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#AiryBiPrimeScaled">AiryBiPrimeScaled</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#AiryBiPrimeScaled">AiryBiPrimeScaled</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#AiryBiScaled">AiryBiScaled</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#AiryBiScaled">AiryBiScaled</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselI">BesselI</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselI">BesselI</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselI0">BesselI0</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselI0MStruveL0">BesselI0MStruveL0</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselI1">BesselI1</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselI1MStruveL1">BesselI1MStruveL1</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselIScaled">BesselIScaled</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselIScaled">BesselIScaled</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselJ">BesselJ</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselJ">BesselJ</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselJScaled">BesselJScaled</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselJScaled">BesselJScaled</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselK">BesselK</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselK">BesselK</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselK0">BesselK0</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselK0e">BesselK0e</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselK1">BesselK1</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselK1e">BesselK1e</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselKScaled">BesselKScaled</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselKScaled">BesselKScaled</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselY">BesselY</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselY">BesselY</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselYScaled">BesselYScaled</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BesselYScaled">BesselYScaled</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#Beta">Beta</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BetaIncomplete">BetaIncomplete</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BetaLn">BetaLn</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BetaRegularized">BetaRegularized</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#Binomial">Binomial</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#BinomialLn">BinomialLn</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#DiGamma">DiGamma</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#DiGammaInv">DiGammaInv</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#Erf">Erf</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#Erfc">Erfc</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#ErfcInv">ErfcInv</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#ErfInv">ErfInv</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#ExponentialIntegral">ExponentialIntegral</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#ExponentialMinusOne">ExponentialMinusOne</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#Factorial">Factorial</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#Factorial">Factorial</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#FactorialLn">FactorialLn</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#FallingFactorial">FallingFactorial</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#Gamma">Gamma</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#GammaLn">GammaLn</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#GammaLowerIncomplete">GammaLowerIncomplete</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#GammaLowerRegularized">GammaLowerRegularized</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#GammaLowerRegularizedInv">GammaLowerRegularizedInv</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#GammaUpperIncomplete">GammaUpperIncomplete</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#GammaUpperRegularized">GammaUpperRegularized</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#GeneralHarmonic">GeneralHarmonic</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#GeneralizedHypergeometric">GeneralizedHypergeometric</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#HankelH1">HankelH1</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#HankelH1Scaled">HankelH1Scaled</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#HankelH2">HankelH2</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#HankelH2Scaled">HankelH2Scaled</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#Harmonic">Harmonic</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#Hypotenuse">Hypotenuse</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#Hypotenuse">Hypotenuse</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#Hypotenuse">Hypotenuse</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#Hypotenuse">Hypotenuse</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#KelvinBe">KelvinBe</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#KelvinBei">KelvinBei</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#KelvinBei">KelvinBei</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#KelvinBeiPrime">KelvinBeiPrime</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#KelvinBeiPrime">KelvinBeiPrime</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#KelvinBer">KelvinBer</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#KelvinBer">KelvinBer</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#KelvinBerPrime">KelvinBerPrime</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#KelvinBerPrime">KelvinBerPrime</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#KelvinKe">KelvinKe</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#KelvinKei">KelvinKei</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#KelvinKei">KelvinKei</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#KelvinKeiPrime">KelvinKeiPrime</a></li>
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<li><a href="../MathNet.Numerics/SpecialFunctions.htm#KelvinKeiPrime">KelvinKeiPrime</a></li>
|
|
<li><a href="../MathNet.Numerics/SpecialFunctions.htm#KelvinKer">KelvinKer</a></li>
|
|
<li><a href="../MathNet.Numerics/SpecialFunctions.htm#KelvinKer">KelvinKer</a></li>
|
|
<li><a href="../MathNet.Numerics/SpecialFunctions.htm#KelvinKerPrime">KelvinKerPrime</a></li>
|
|
<li><a href="../MathNet.Numerics/SpecialFunctions.htm#KelvinKerPrime">KelvinKerPrime</a></li>
|
|
<li><a href="../MathNet.Numerics/SpecialFunctions.htm#Logistic">Logistic</a></li>
|
|
<li><a href="../MathNet.Numerics/SpecialFunctions.htm#Logit">Logit</a></li>
|
|
<li><a href="../MathNet.Numerics/SpecialFunctions.htm#MarcumQ">MarcumQ</a></li>
|
|
<li><a href="../MathNet.Numerics/SpecialFunctions.htm#MarcumQ">MarcumQ</a></li>
|
|
<li><a href="../MathNet.Numerics/SpecialFunctions.htm#Multinomial">Multinomial</a></li>
|
|
<li><a href="../MathNet.Numerics/SpecialFunctions.htm#RisingFactorial">RisingFactorial</a></li>
|
|
<li><a href="../MathNet.Numerics/SpecialFunctions.htm#SphericalBesselJ">SphericalBesselJ</a></li>
|
|
<li><a href="../MathNet.Numerics/SpecialFunctions.htm#SphericalBesselJ">SphericalBesselJ</a></li>
|
|
<li><a href="../MathNet.Numerics/SpecialFunctions.htm#SphericalBesselY">SphericalBesselY</a></li>
|
|
<li><a href="../MathNet.Numerics/SpecialFunctions.htm#SphericalBesselY">SphericalBesselY</a></li>
|
|
<li><a href="../MathNet.Numerics/SpecialFunctions.htm#StruveL0">StruveL0</a></li>
|
|
<li><a href="../MathNet.Numerics/SpecialFunctions.htm#StruveL1">StruveL1</a></li>
|
|
</ul>
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
<h3 class="section">Public Static Functions</h3>
|
|
|
|
<div id="AiryAi" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>AiryAi</strong>(<span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the Airy function Ai. <p>AiryAi(z) is a solution to the Airy equation, y'' - y * z = 0. </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the Airy function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The Airy function Ai. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="AiryAi" class="method">
|
|
<h4><span title="System.double">double</span> <strong>AiryAi</strong>(<span title="System.double">double</span> z)</h4>
|
|
<div class="content">Returns the Airy function Ai. <p>AiryAi(z) is a solution to the Airy equation, y'' - y * z = 0. </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The value to compute the Airy function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The Airy function Ai. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="AiryAiPrime" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>AiryAiPrime</strong>(<span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the derivative of the Airy function Ai. <p>AiryAiPrime(z) is defined as d/dz AiryAi(z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the derivative of the Airy function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The derivative of the Airy function Ai. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="AiryAiPrime" class="method">
|
|
<h4><span title="System.double">double</span> <strong>AiryAiPrime</strong>(<span title="System.double">double</span> z)</h4>
|
|
<div class="content">Returns the derivative of the Airy function Ai. <p>AiryAiPrime(z) is defined as d/dz AiryAi(z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The value to compute the derivative of the Airy function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The derivative of the Airy function Ai. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="AiryAiPrimeScaled" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>AiryAiPrimeScaled</strong>(<span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the exponentially scaled derivative of Airy function Ai <p>ScaledAiryAiPrime(z) is given by Exp(zta) * AiryAiPrime(z), where zta = (2/3) * z * Sqrt(z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the derivative of the Airy function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The exponentially scaled derivative of Airy function Ai. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="AiryAiPrimeScaled" class="method">
|
|
<h4><span title="System.double">double</span> <strong>AiryAiPrimeScaled</strong>(<span title="System.double">double</span> z)</h4>
|
|
<div class="content">Returns the exponentially scaled derivative of the Airy function Ai. <p>ScaledAiryAiPrime(z) is given by Exp(zta) * AiryAiPrime(z), where zta = (2/3) * z * Sqrt(z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The value to compute the derivative of the Airy function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The exponentially scaled derivative of the Airy function Ai. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="AiryAiScaled" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>AiryAiScaled</strong>(<span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the exponentially scaled Airy function Ai. <p>ScaledAiryAi(z) is given by Exp(zta) * AiryAi(z), where zta = (2/3) * z * Sqrt(z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the Airy function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The exponentially scaled Airy function Ai. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="AiryAiScaled" class="method">
|
|
<h4><span title="System.double">double</span> <strong>AiryAiScaled</strong>(<span title="System.double">double</span> z)</h4>
|
|
<div class="content">Returns the exponentially scaled Airy function Ai. <p>ScaledAiryAi(z) is given by Exp(zta) * AiryAi(z), where zta = (2/3) * z * Sqrt(z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The value to compute the Airy function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The exponentially scaled Airy function Ai. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="AiryBi" class="method">
|
|
<h4><span title="System.double">double</span> <strong>AiryBi</strong>(<span title="System.double">double</span> z)</h4>
|
|
<div class="content">Returns the Airy function Bi. <p>AiryBi(z) is a solution to the Airy equation, y'' - y * z = 0. </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The value to compute the Airy function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The Airy function Bi. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="AiryBi" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>AiryBi</strong>(<span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the Airy function Bi. <p>AiryBi(z) is a solution to the Airy equation, y'' - y * z = 0. </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the Airy function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The Airy function Bi. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="AiryBiPrime" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>AiryBiPrime</strong>(<span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the derivative of the Airy function Bi. <p>AiryBiPrime(z) is defined as d/dz AiryBi(z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the derivative of the Airy function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The derivative of the Airy function Bi. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="AiryBiPrime" class="method">
|
|
<h4><span title="System.double">double</span> <strong>AiryBiPrime</strong>(<span title="System.double">double</span> z)</h4>
|
|
<div class="content">Returns the derivative of the Airy function Bi. <p>AiryBiPrime(z) is defined as d/dz AiryBi(z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The value to compute the derivative of the Airy function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The derivative of the Airy function Bi. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="AiryBiPrimeScaled" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>AiryBiPrimeScaled</strong>(<span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the exponentially scaled derivative of the Airy function Bi. <p>ScaledAiryBiPrime(z) is given by Exp(-Abs(zta.Real)) * AiryBiPrime(z) where zta = (2 / 3) * z * Sqrt(z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the derivative of the Airy function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The exponentially scaled derivative of the Airy function Bi. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="AiryBiPrimeScaled" class="method">
|
|
<h4><span title="System.double">double</span> <strong>AiryBiPrimeScaled</strong>(<span title="System.double">double</span> z)</h4>
|
|
<div class="content">Returns the exponentially scaled derivative of the Airy function Bi. <p>ScaledAiryBiPrime(z) is given by Exp(-Abs(zta.Real)) * AiryBiPrime(z) where zta = (2 / 3) * z * Sqrt(z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The value to compute the derivative of the Airy function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The exponentially scaled derivative of the Airy function Bi. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="AiryBiScaled" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>AiryBiScaled</strong>(<span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the exponentially scaled Airy function Bi. <p>ScaledAiryBi(z) is given by Exp(-Abs(zta.Real)) * AiryBi(z) where zta = (2 / 3) * z * Sqrt(z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the Airy function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The exponentially scaled Airy function Bi(z). </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="AiryBiScaled" class="method">
|
|
<h4><span title="System.double">double</span> <strong>AiryBiScaled</strong>(<span title="System.double">double</span> z)</h4>
|
|
<div class="content">Returns the exponentially scaled Airy function Bi. <p>ScaledAiryBi(z) is given by Exp(-Abs(zta.Real)) * AiryBi(z) where zta = (2 / 3) * z * Sqrt(z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The value to compute the Airy function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The exponentially scaled Airy function Bi. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselI" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BesselI</strong>(<span title="System.double">double</span> n, <span title="System.double">double</span> z)</h4>
|
|
<div class="content">Returns the modified Bessel function of the first kind. <p>BesselI(n, z) is a solution to the modified Bessel differential equation. </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the modified Bessel function. </p>
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The value to compute the modified Bessel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The modified Bessel function of the first kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselI" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>BesselI</strong>(<span title="System.double">double</span> n, <span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the modified Bessel function of the first kind. <p>BesselI(n, z) is a solution to the modified Bessel differential equation. </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the modified Bessel function. </p>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the modified Bessel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The modified Bessel function of the first kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselI0" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BesselI0</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the modified Bessel function of first kind, order 0 of the argument. The function is defined as . The range is partitioned into the two intervals [0, 8] and
|
|
(8, infinity). Chebyshev polynomial expansions are employed
|
|
in each interval.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the Bessel function of. </p>
|
|
</div>
|
|
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselI0MStruveL0" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BesselI0MStruveL0</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the difference between the Bessel I0 and Struve L0 functions.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the function of. </p>
|
|
</div>
|
|
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselI1" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BesselI1</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the modified Bessel function of first kind,
|
|
order 1 of the argument. The function is defined as . The range is partitioned into the two intervals [0, 8] and
|
|
(8, infinity). Chebyshev polynomial expansions are employed
|
|
in each interval.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the Bessel function of. </p>
|
|
</div>
|
|
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselI1MStruveL1" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BesselI1MStruveL1</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the difference between the Bessel I1 and Struve L1 functions.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the function of. </p>
|
|
</div>
|
|
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselIScaled" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>BesselIScaled</strong>(<span title="System.double">double</span> n, <span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the exponentially scaled modified Bessel function of the first kind. <p>ScaledBesselI(n, z) is given by Exp(-Abs(z.Real)) * BesselI(n, z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the modified Bessel function. </p>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the modified Bessel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The exponentially scaled modified Bessel function of the first kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselIScaled" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BesselIScaled</strong>(<span title="System.double">double</span> n, <span title="System.double">double</span> z)</h4>
|
|
<div class="content">Returns the exponentially scaled modified Bessel function of the first kind. <p>ScaledBesselI(n, z) is given by Exp(-Abs(z.Real)) * BesselI(n, z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the modified Bessel function. </p>
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The value to compute the modified Bessel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The exponentially scaled modified Bessel function of the first kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselJ" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BesselJ</strong>(<span title="System.double">double</span> n, <span title="System.double">double</span> z)</h4>
|
|
<div class="content">Returns the Bessel function of the first kind. <p>BesselJ(n, z) is a solution to the Bessel differential equation. </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the Bessel function. </p>
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The value to compute the Bessel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The Bessel function of the first kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselJ" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>BesselJ</strong>(<span title="System.double">double</span> n, <span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the Bessel function of the first kind. <p>BesselJ(n, z) is a solution to the Bessel differential equation. </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the Bessel function. </p>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the Bessel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The Bessel function of the first kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselJScaled" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BesselJScaled</strong>(<span title="System.double">double</span> n, <span title="System.double">double</span> z)</h4>
|
|
<div class="content">Returns the exponentially scaled Bessel function of the first kind. <p>ScaledBesselJ(n, z) is given by Exp(-Abs(z.Imaginary)) * BesselJ(n, z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the Bessel function. </p>
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The value to compute the Bessel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The exponentially scaled Bessel function of the first kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselJScaled" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>BesselJScaled</strong>(<span title="System.double">double</span> n, <span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the exponentially scaled Bessel function of the first kind. <p>ScaledBesselJ(n, z) is given by Exp(-Abs(z.Imaginary)) * BesselJ(n, z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the Bessel function. </p>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the Bessel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The exponentially scaled Bessel function of the first kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselK" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>BesselK</strong>(<span title="System.double">double</span> n, <span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the modified Bessel function of the second kind. <p>BesselK(n, z) is a solution to the modified Bessel differential equation. </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the modified Bessel function. </p>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the modified Bessel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The modified Bessel function of the second kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselK" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BesselK</strong>(<span title="System.double">double</span> n, <span title="System.double">double</span> z)</h4>
|
|
<div class="content">Returns the modified Bessel function of the second kind. <p>BesselK(n, z) is a solution to the modified Bessel differential equation. </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the modified Bessel function. </p>
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The value to compute the modified Bessel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The modified Bessel function of the second kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselK0" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BesselK0</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the modified Bessel function of the second kind
|
|
of order 0 of the argument. The range is partitioned into the two intervals [0, 8] and
|
|
(8, infinity). Chebyshev polynomial expansions are employed
|
|
in each interval.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the Bessel function of. </p>
|
|
</div>
|
|
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselK0e" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BesselK0e</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the exponentially scaled modified Bessel function
|
|
of the second kind of order 0 of the argument.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the Bessel function of. </p>
|
|
</div>
|
|
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselK1" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BesselK1</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the modified Bessel function of the second kind
|
|
of order 1 of the argument. The range is partitioned into the two intervals [0, 2] and
|
|
(2, infinity). Chebyshev polynomial expansions are employed
|
|
in each interval.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the Bessel function of. </p>
|
|
</div>
|
|
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselK1e" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BesselK1e</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the exponentially scaled modified Bessel function
|
|
of the second kind of order 1 of the argument..
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the Bessel function of. </p>
|
|
</div>
|
|
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselKScaled" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>BesselKScaled</strong>(<span title="System.double">double</span> n, <span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the exponentially scaled modified Bessel function of the second kind. <p>ScaledBesselK(n, z) is given by Exp(z) * BesselK(n, z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the modified Bessel function. </p>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the modified Bessel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The exponentially scaled modified Bessel function of the second kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselKScaled" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BesselKScaled</strong>(<span title="System.double">double</span> n, <span title="System.double">double</span> z)</h4>
|
|
<div class="content">Returns the exponentially scaled modified Bessel function of the second kind. <p>ScaledBesselK(n, z) is given by Exp(z) * BesselK(n, z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the modified Bessel function. </p>
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The value to compute the modified Bessel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The exponentially scaled modified Bessel function of the second kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselY" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BesselY</strong>(<span title="System.double">double</span> n, <span title="System.double">double</span> z)</h4>
|
|
<div class="content">Returns the Bessel function of the second kind. <p>BesselY(n, z) is a solution to the Bessel differential equation. </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the Bessel function. </p>
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The value to compute the Bessel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The Bessel function of the second kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselY" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>BesselY</strong>(<span title="System.double">double</span> n, <span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the Bessel function of the second kind. <p>BesselY(n, z) is a solution to the Bessel differential equation. </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the Bessel function. </p>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the Bessel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The Bessel function of the second kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselYScaled" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BesselYScaled</strong>(<span title="System.double">double</span> n, <span title="System.double">double</span> z)</h4>
|
|
<div class="content">Returns the exponentially scaled Bessel function of the second kind. <p>ScaledBesselY(n, z) is given by Exp(-Abs(z.Imaginary)) * BesselY(n, z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the Bessel function. </p>
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The value to compute the Bessel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The exponentially scaled Bessel function of the second kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BesselYScaled" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>BesselYScaled</strong>(<span title="System.double">double</span> n, <span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the exponentially scaled Bessel function of the second kind. <p>ScaledBesselY(n, z) is given by Exp(-Abs(z.Imaginary)) * Y(n, z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the Bessel function. </p>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the Bessel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The exponentially scaled Bessel function of the second kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="Beta" class="method">
|
|
<h4><span title="System.double">double</span> <strong>Beta</strong>(<span title="System.double">double</span> z, <span title="System.double">double</span> w)</h4>
|
|
<div class="content">Computes the Euler Beta function.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The first Beta parameter, a positive real number. </p>
|
|
<h6><code><span title="System.double">double</span></code> w</h6>
|
|
<p class="comments">The second Beta parameter, a positive real number. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The Euler Beta function evaluated at z,w. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BetaIncomplete" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BetaIncomplete</strong>(<span title="System.double">double</span> a, <span title="System.double">double</span> b, <span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the lower incomplete (unregularized) beta function
|
|
B(a,b,x) = int(t^(a-1)*(1-t)^(b-1),t=0..x) for real a > 0, b > 0, 1 >= x >= 0.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> a</h6>
|
|
<p class="comments">The first Beta parameter, a positive real number. </p>
|
|
<h6><code><span title="System.double">double</span></code> b</h6>
|
|
<p class="comments">The second Beta parameter, a positive real number. </p>
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The upper limit of the integral. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The lower incomplete (unregularized) beta function. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BetaLn" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BetaLn</strong>(<span title="System.double">double</span> z, <span title="System.double">double</span> w)</h4>
|
|
<div class="content">Computes the logarithm of the Euler Beta function.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The first Beta parameter, a positive real number. </p>
|
|
<h6><code><span title="System.double">double</span></code> w</h6>
|
|
<p class="comments">The second Beta parameter, a positive real number. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The logarithm of the Euler Beta function evaluated at z,w. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BetaRegularized" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BetaRegularized</strong>(<span title="System.double">double</span> a, <span title="System.double">double</span> b, <span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the regularized lower incomplete beta function
|
|
I_x(a,b) = 1/Beta(a,b) * int(t^(a-1)*(1-t)^(b-1),t=0..x) for real a > 0, b > 0, 1 >= x >= 0.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> a</h6>
|
|
<p class="comments">The first Beta parameter, a positive real number. </p>
|
|
<h6><code><span title="System.double">double</span></code> b</h6>
|
|
<p class="comments">The second Beta parameter, a positive real number. </p>
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The upper limit of the integral. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The regularized lower incomplete beta function. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="Binomial" class="method">
|
|
<h4><span title="System.double">double</span> <strong>Binomial</strong>(<span title="System.int">int</span> n, <span title="System.int">int</span> k)</h4>
|
|
<div class="content">Computes the binomial coefficient: n choose k.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.int">int</span></code> n</h6>
|
|
<p class="comments">A nonnegative value n. </p>
|
|
<h6><code><span title="System.int">int</span></code> k</h6>
|
|
<p class="comments">A nonnegative value h. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The binomial coefficient: n choose k. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="BinomialLn" class="method">
|
|
<h4><span title="System.double">double</span> <strong>BinomialLn</strong>(<span title="System.int">int</span> n, <span title="System.int">int</span> k)</h4>
|
|
<div class="content">Computes the natural logarithm of the binomial coefficient: ln(n choose k).
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.int">int</span></code> n</h6>
|
|
<p class="comments">A nonnegative value n. </p>
|
|
<h6><code><span title="System.int">int</span></code> k</h6>
|
|
<p class="comments">A nonnegative value h. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The logarithmic binomial coefficient: ln(n choose k). </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="DiGamma" class="method">
|
|
<h4><span title="System.double">double</span> <strong>DiGamma</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Computes the Digamma function which is mathematically defined as the derivative of the logarithm of the gamma function.
|
|
This implementation is based on
|
|
Jose Bernardo
|
|
Algorithm AS 103:
|
|
Psi ( Digamma ) Function,
|
|
Applied Statistics,
|
|
Volume 25, Number 3, 1976, pages 315-317.
|
|
Using the modifications as in Tom Minka's lightspeed toolbox.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The argument of the digamma function. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The value of the DiGamma function at <var>x</var>. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="DiGammaInv" class="method">
|
|
<h4><span title="System.double">double</span> <strong>DiGammaInv</strong>(<span title="System.double">double</span> p)</h4>
|
|
<div class="content"><p>Computes the inverse Digamma function: this is the inverse of the logarithm of the gamma function. This function will
|
|
only return solutions that are positive. </p> <p>This implementation is based on the bisection method. </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> p</h6>
|
|
<p class="comments">The argument of the inverse digamma function. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The positive solution to the inverse DiGamma function at <var>p</var>. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="Erf" class="method">
|
|
<h4><span title="System.double">double</span> <strong>Erf</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Calculates the error function. <blockquote class="remarks">
|
|
<ul><li></li><li></li></ul>
|
|
</blockquote>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to evaluate. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>the error function evaluated at given value. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="Erfc" class="method">
|
|
<h4><span title="System.double">double</span> <strong>Erfc</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Calculates the complementary error function. <blockquote class="remarks">
|
|
<ul><li></li><li></li></ul>
|
|
</blockquote>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to evaluate. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>the complementary error function evaluated at given value. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="ErfcInv" class="method">
|
|
<h4><span title="System.double">double</span> <strong>ErfcInv</strong>(<span title="System.double">double</span> z)</h4>
|
|
<div class="content">Calculates the complementary inverse error function evaluated at z. <blockquote class="remarks">
|
|
We have tested this implementation against the arbitrary precision mpmath library
|
|
and found cases where we can only guarantee 9 significant figures correct. <ul><li></li><li></li></ul>
|
|
</blockquote>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">value to evaluate. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The complementary inverse error function evaluated at given value. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="ErfInv" class="method">
|
|
<h4><span title="System.double">double</span> <strong>ErfInv</strong>(<span title="System.double">double</span> z)</h4>
|
|
<div class="content">Calculates the inverse error function evaluated at z. <blockquote class="remarks">
|
|
<ul><li></li><li></li></ul>
|
|
</blockquote>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">value to evaluate. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The inverse error function evaluated at given value. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="ExponentialIntegral" class="method">
|
|
<h4><span title="System.double">double</span> <strong>ExponentialIntegral</strong>(<span title="System.double">double</span> x, <span title="System.int">int</span> n)</h4>
|
|
<div class="content">Computes the generalized Exponential Integral function (En). <blockquote class="remarks">
|
|
<p>This implementation of the computation of the Exponential Integral function follows the derivation in
|
|
"Handbook of Mathematical Functions, Applied Mathematics Series, Volume 55", Abramowitz, M., and Stegun, I.A. 1964, reprinted 1968 by
|
|
Dover Publications, New York), Chapters 6, 7, and 26.
|
|
AND
|
|
"Advanced mathematical methods for scientists and engineers", Bender, Carl M.; Steven A. Orszag (1978). page 253 </p> <p>for x > 1 uses continued fraction approach that is often used to compute incomplete gamma.
|
|
for 0 < x <= 1 uses Taylor series expansion </p> <p>Our unit tests suggest that the accuracy of the Exponential Integral function is correct up to 13 floating point digits. </p>
|
|
</blockquote>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The argument of the Exponential Integral function. </p>
|
|
<h6><code><span title="System.int">int</span></code> n</h6>
|
|
<p class="comments">Integer power of the denominator term. Generalization index. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The value of the Exponential Integral function. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="ExponentialMinusOne" class="method">
|
|
<h4><span title="System.double">double</span> <strong>ExponentialMinusOne</strong>(<span title="System.double">double</span> power)</h4>
|
|
<div class="content">Numerically stable exponential minus one, i.e. <code>x -> exp(x)-1</code>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> power</h6>
|
|
<p class="comments">A number specifying a power. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>Returns <code>exp(power)-1</code>. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="Factorial" class="method">
|
|
<h4><span title="System.Numerics.BigInteger">BigInteger</span> <strong>Factorial</strong>(<span title="System.Numerics.BigInteger">BigInteger</span> x)</h4>
|
|
<div class="content">Computes the factorial of an integer.
|
|
|
|
|
|
|
|
|
|
</div>
|
|
</div>
|
|
<div id="Factorial" class="method">
|
|
<h4><span title="System.double">double</span> <strong>Factorial</strong>(<span title="System.int">int</span> x)</h4>
|
|
<div class="content">Computes the factorial function x -> x! of an integer number > 0. The function can represent all number up
|
|
to 22! exactly, all numbers up to 170! using a double representation. All larger values will overflow. <blockquote class="remarks">
|
|
If you need to multiply or divide various such factorials, consider using the logarithmic version <a href="../MathNet.Numerics/SpecialFunctions.htm#FactorialLn">FactorialLn</a> instead so you can add instead of multiply and subtract instead of divide, and
|
|
then exponentiate the result using <span title="Exp">Exp</span>. This will also circumvent the problem that
|
|
factorials become very large even for small parameters.
|
|
</blockquote>
|
|
|
|
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>A value value! for value > 0 </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="FactorialLn" class="method">
|
|
<h4><span title="System.double">double</span> <strong>FactorialLn</strong>(<span title="System.int">int</span> x)</h4>
|
|
<div class="content">Computes the logarithmic factorial function x -> ln(x!) of an integer number > 0.
|
|
|
|
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>A value value! for value > 0 </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="FallingFactorial" class="method">
|
|
<h4><span title="System.double">double</span> <strong>FallingFactorial</strong>(<span title="System.double">double</span> x, <span title="System.int">int</span> n)</h4>
|
|
<div class="content">Computes the Falling Factorial (Pochhammer function) x -> x(n), n>= 0. see: https://en.wikipedia.org/wiki/Falling_and_rising_factorials
|
|
|
|
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The real value of the Falling Factorial for x and n </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="Gamma" class="method">
|
|
<h4><span title="System.double">double</span> <strong>Gamma</strong>(<span title="System.double">double</span> z)</h4>
|
|
<div class="content">Computes the Gamma function. <blockquote class="remarks">
|
|
<p>This implementation of the computation of the gamma and logarithm of the gamma function follows the derivation in
|
|
"An Analysis Of The Lanczos Gamma Approximation", Glendon Ralph Pugh, 2004.
|
|
We use the implementation listed on p. 116 which should achieve an accuracy of 16 floating point digits. Although 16 digit accuracy
|
|
should be sufficient for double values, improving accuracy is possible (see p. 126 in Pugh). </p> <p>Our unit tests suggest that the accuracy of the Gamma function is correct up to 13 floating point digits. </p>
|
|
</blockquote>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The argument of the gamma function. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The logarithm of the gamma function. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="GammaLn" class="method">
|
|
<h4><span title="System.double">double</span> <strong>GammaLn</strong>(<span title="System.double">double</span> z)</h4>
|
|
<div class="content">Computes the logarithm of the Gamma function. <blockquote class="remarks">
|
|
<p>This implementation of the computation of the gamma and logarithm of the gamma function follows the derivation in
|
|
"An Analysis Of The Lanczos Gamma Approximation", Glendon Ralph Pugh, 2004.
|
|
We use the implementation listed on p. 116 which achieves an accuracy of 16 floating point digits. Although 16 digit accuracy
|
|
should be sufficient for double values, improving accuracy is possible (see p. 126 in Pugh). </p> <p>Our unit tests suggest that the accuracy of the Gamma function is correct up to 14 floating point digits. </p>
|
|
</blockquote>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The argument of the gamma function. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The logarithm of the gamma function. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="GammaLowerIncomplete" class="method">
|
|
<h4><span title="System.double">double</span> <strong>GammaLowerIncomplete</strong>(<span title="System.double">double</span> a, <span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the lower incomplete gamma function
|
|
gamma(a,x) = int(exp(-t)t^(a-1),t=0..x) for real a > 0, x > 0.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> a</h6>
|
|
<p class="comments">The argument for the gamma function. </p>
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The upper integral limit. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The lower incomplete gamma function. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="GammaLowerRegularized" class="method">
|
|
<h4><span title="System.double">double</span> <strong>GammaLowerRegularized</strong>(<span title="System.double">double</span> a, <span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the lower incomplete regularized gamma function
|
|
P(a,x) = 1/Gamma(a) * int(exp(-t)t^(a-1),t=0..x) for real a > 0, x > 0.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> a</h6>
|
|
<p class="comments">The argument for the gamma function. </p>
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The upper integral limit. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The lower incomplete gamma function. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="GammaLowerRegularizedInv" class="method">
|
|
<h4><span title="System.double">double</span> <strong>GammaLowerRegularizedInv</strong>(<span title="System.double">double</span> a, <span title="System.double">double</span> y0)</h4>
|
|
<div class="content">Returns the inverse P^(-1) of the regularized lower incomplete gamma function
|
|
P(a,x) = 1/Gamma(a) * int(exp(-t)t^(a-1),t=0..x) for real a > 0, x > 0,
|
|
such that P^(-1)(a,P(a,x)) == x.
|
|
|
|
|
|
|
|
|
|
</div>
|
|
</div>
|
|
<div id="GammaUpperIncomplete" class="method">
|
|
<h4><span title="System.double">double</span> <strong>GammaUpperIncomplete</strong>(<span title="System.double">double</span> a, <span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the upper incomplete gamma function
|
|
Gamma(a,x) = int(exp(-t)t^(a-1),t=0..x) for real a > 0, x > 0.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> a</h6>
|
|
<p class="comments">The argument for the gamma function. </p>
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The lower integral limit. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The upper incomplete gamma function. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="GammaUpperRegularized" class="method">
|
|
<h4><span title="System.double">double</span> <strong>GammaUpperRegularized</strong>(<span title="System.double">double</span> a, <span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the upper incomplete regularized gamma function
|
|
Q(a,x) = 1/Gamma(a) * int(exp(-t)t^(a-1),t=0..x) for real a > 0, x > 0.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> a</h6>
|
|
<p class="comments">The argument for the gamma function. </p>
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The lower integral limit. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The upper incomplete regularized gamma function. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="GeneralHarmonic" class="method">
|
|
<h4><span title="System.double">double</span> <strong>GeneralHarmonic</strong>(<span title="System.int">int</span> n, <span title="System.double">double</span> m)</h4>
|
|
<div class="content">Compute the generalized harmonic number of order n of m. (1 + 1/2^m + 1/3^m +... + 1/n^m)
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.int">int</span></code> n</h6>
|
|
<p class="comments">The order parameter. </p>
|
|
<h6><code><span title="System.double">double</span></code> m</h6>
|
|
<p class="comments">The power parameter. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>General Harmonic number. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="GeneralizedHypergeometric" class="method">
|
|
<h4><span title="System.double">double</span> <strong>GeneralizedHypergeometric</strong>(<span title="System.Double[]">Double[]</span> a, <span title="System.Double[]">Double[]</span> b, <span title="System.int">int</span> z)</h4>
|
|
<div class="content">A generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n.
|
|
This is the most common pFq(a1,..., ap; b1,...,bq; z) representation
|
|
see: https://en.wikipedia.org/wiki/Generalized_hypergeometric_function
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.Double[]">Double[]</span></code> a</h6>
|
|
<p class="comments">The list of coefficients in the numerator </p>
|
|
<h6><code><span title="System.Double[]">Double[]</span></code> b</h6>
|
|
<p class="comments">The list of coefficients in the denominator </p>
|
|
<h6><code><span title="System.int">int</span></code> z</h6>
|
|
<p class="comments">The variable in the power series </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The value of the Generalized HyperGeometric Function. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="HankelH1" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>HankelH1</strong>(<span title="System.double">double</span> n, <span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the Hankel function of the first kind. <p>HankelH1(n, z) is defined as BesselJ(n, z) + j * BesselY(n, z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the Hankel function. </p>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the Hankel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The Hankel function of the first kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="HankelH1Scaled" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>HankelH1Scaled</strong>(<span title="System.double">double</span> n, <span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the exponentially scaled Hankel function of the first kind. <p>ScaledHankelH1(n, z) is given by Exp(-z * j) * HankelH1(n, z) where j = Sqrt(-1). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the Hankel function. </p>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the Hankel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The exponentially scaled Hankel function of the first kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="HankelH2" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>HankelH2</strong>(<span title="System.double">double</span> n, <span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the Hankel function of the second kind. <p>HankelH2(n, z) is defined as BesselJ(n, z) - j * BesselY(n, z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the Hankel function. </p>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the Hankel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The Hankel function of the second kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="HankelH2Scaled" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>HankelH2Scaled</strong>(<span title="System.double">double</span> n, <span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the exponentially scaled Hankel function of the second kind. <p>ScaledHankelH2(n, z) is given by Exp(z * j) * HankelH2(n, z) where j = Sqrt(-1). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the Hankel function. </p>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the Hankel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The exponentially scaled Hankel function of the second kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="Harmonic" class="method">
|
|
<h4><span title="System.double">double</span> <strong>Harmonic</strong>(<span title="System.int">int</span> t)</h4>
|
|
<div class="content">Computes the <var>t</var> 'th Harmonic number.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.int">int</span></code> t</h6>
|
|
<p class="comments">The Harmonic number which needs to be computed. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The t'th Harmonic number. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="Hypotenuse" class="method">
|
|
<h4><span title="System.float">float</span> <strong>Hypotenuse</strong>(<span title="System.float">float</span> a, <span title="System.float">float</span> b)</h4>
|
|
<div class="content">Numerically stable hypotenuse of a right angle triangle, i.e. <code>(a,b) -> sqrt(a^2 + b^2)</code>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.float">float</span></code> a</h6>
|
|
<p class="comments">The length of side a of the triangle. </p>
|
|
<h6><code><span title="System.float">float</span></code> b</h6>
|
|
<p class="comments">The length of side b of the triangle. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.float">float</span></code></h6>
|
|
<p>Returns <code>sqrt(a2 + b2)</code> without underflow/overflow. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="Hypotenuse" class="method">
|
|
<h4><span title="System.double">double</span> <strong>Hypotenuse</strong>(<span title="System.double">double</span> a, <span title="System.double">double</span> b)</h4>
|
|
<div class="content">Numerically stable hypotenuse of a right angle triangle, i.e. <code>(a,b) -> sqrt(a^2 + b^2)</code>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> a</h6>
|
|
<p class="comments">The length of side a of the triangle. </p>
|
|
<h6><code><span title="System.double">double</span></code> b</h6>
|
|
<p class="comments">The length of side b of the triangle. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>Returns <code>sqrt(a2 + b2)</code> without underflow/overflow. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="Hypotenuse" class="method">
|
|
<h4><a href="../MathNet.Numerics/Complex32.htm">Complex32</a> <strong>Hypotenuse</strong>(<a href="../MathNet.Numerics/Complex32.htm">Complex32</a> a, <a href="../MathNet.Numerics/Complex32.htm">Complex32</a> b)</h4>
|
|
<div class="content">Numerically stable hypotenuse of a right angle triangle, i.e. <code>(a,b) -> sqrt(a^2 + b^2)</code>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><a href="../MathNet.Numerics/Complex32.htm">Complex32</a></code> a</h6>
|
|
<p class="comments">The length of side a of the triangle. </p>
|
|
<h6><code><a href="../MathNet.Numerics/Complex32.htm">Complex32</a></code> b</h6>
|
|
<p class="comments">The length of side b of the triangle. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><a href="../MathNet.Numerics/Complex32.htm">Complex32</a></code></h6>
|
|
<p>Returns <code>sqrt(a2 + b2)</code> without underflow/overflow. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="Hypotenuse" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>Hypotenuse</strong>(<span title="System.Numerics.Complex">Complex</span> a, <span title="System.Numerics.Complex">Complex</span> b)</h4>
|
|
<div class="content">Numerically stable hypotenuse of a right angle triangle, i.e. <code>(a,b) -> sqrt(a^2 + b^2)</code>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> a</h6>
|
|
<p class="comments">The length of side a of the triangle. </p>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> b</h6>
|
|
<p class="comments">The length of side b of the triangle. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>Returns <code>sqrt(a2 + b2)</code> without underflow/overflow. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="KelvinBe" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>KelvinBe</strong>(<span title="System.double">double</span> nu, <span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the Kelvin function of the first kind. <p>KelvinBe(nu, x) is given by BesselJ(0, j * sqrt(j) * x) where j = sqrt(-1). </p> <p>KelvinBer(nu, x) and KelvinBei(nu, x) are the real and imaginary parts of the KelvinBe(nu, x) </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> nu</h6>
|
|
<p class="comments">the order of the the Kelvin function. </p>
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the Kelvin function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The Kelvin function of the first kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="KelvinBei" class="method">
|
|
<h4><span title="System.double">double</span> <strong>KelvinBei</strong>(<span title="System.double">double</span> nu, <span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the Kelvin function bei. <p>KelvinBei(nu, x) is given by the imaginary part of BesselJ(nu, j * sqrt(j) * x) where j = sqrt(-1). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> nu</h6>
|
|
<p class="comments">the order of the the Kelvin function. </p>
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the Kelvin function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The Kelvin function bei. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="KelvinBei" class="method">
|
|
<h4><span title="System.double">double</span> <strong>KelvinBei</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the Kelvin function bei. <p>KelvinBei(x) is given by the imaginary part of BesselJ(0, j * sqrt(j) * x) where j = sqrt(-1). </p> <p>KelvinBei(x) is equivalent to KelvinBei(0, x). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the Kelvin function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The Kelvin function bei. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="KelvinBeiPrime" class="method">
|
|
<h4><span title="System.double">double</span> <strong>KelvinBeiPrime</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the derivative of the Kelvin function bei.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the derivative of the Kelvin function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The derivative of the Kelvin function bei. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="KelvinBeiPrime" class="method">
|
|
<h4><span title="System.double">double</span> <strong>KelvinBeiPrime</strong>(<span title="System.double">double</span> nu, <span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the derivative of the Kelvin function bei.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> nu</h6>
|
|
<p class="comments">The order of the Kelvin function. </p>
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the derivative of the Kelvin function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>the derivative of the Kelvin function bei. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="KelvinBer" class="method">
|
|
<h4><span title="System.double">double</span> <strong>KelvinBer</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the Kelvin function ber. <p>KelvinBer(x) is given by the real part of BesselJ(0, j * sqrt(j) * x) where j = sqrt(-1). </p> <p>KelvinBer(x) is equivalent to KelvinBer(0, x). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the Kelvin function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The Kelvin function ber. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="KelvinBer" class="method">
|
|
<h4><span title="System.double">double</span> <strong>KelvinBer</strong>(<span title="System.double">double</span> nu, <span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the Kelvin function ber. <p>KelvinBer(nu, x) is given by the real part of BesselJ(nu, j * sqrt(j) * x) where j = sqrt(-1). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> nu</h6>
|
|
<p class="comments">the order of the the Kelvin function. </p>
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the Kelvin function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The Kelvin function ber. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="KelvinBerPrime" class="method">
|
|
<h4><span title="System.double">double</span> <strong>KelvinBerPrime</strong>(<span title="System.double">double</span> nu, <span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the derivative of the Kelvin function ber.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> nu</h6>
|
|
<p class="comments">The order of the Kelvin function. </p>
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the derivative of the Kelvin function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>the derivative of the Kelvin function ber </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="KelvinBerPrime" class="method">
|
|
<h4><span title="System.double">double</span> <strong>KelvinBerPrime</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the derivative of the Kelvin function ber.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the derivative of the Kelvin function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The derivative of the Kelvin function ber. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="KelvinKe" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>KelvinKe</strong>(<span title="System.double">double</span> nu, <span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the Kelvin function of the second kind <p>KelvinKe(nu, x) is given by Exp(-nu * pi * j / 2) * BesselK(nu, x * sqrt(j)) where j = sqrt(-1). </p> <p>KelvinKer(nu, x) and KelvinKei(nu, x) are the real and imaginary parts of the KelvinBe(nu, x) </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> nu</h6>
|
|
<p class="comments">The order of the Kelvin function. </p>
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to calculate the kelvin function of, </p>
|
|
</div>
|
|
|
|
|
|
</div>
|
|
</div>
|
|
<div id="KelvinKei" class="method">
|
|
<h4><span title="System.double">double</span> <strong>KelvinKei</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the Kelvin function kei. <p>KelvinKei(x) is given by the imaginary part of Exp(-nu * pi * j / 2) * BesselK(0, sqrt(j) * x) where j = sqrt(-1). </p> <p>KelvinKei(x) is equivalent to KelvinKei(0, x). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The non-negative real value to compute the Kelvin function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The Kelvin function kei. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="KelvinKei" class="method">
|
|
<h4><span title="System.double">double</span> <strong>KelvinKei</strong>(<span title="System.double">double</span> nu, <span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the Kelvin function kei. <p>KelvinKei(nu, x) is given by the imaginary part of Exp(-nu * pi * j / 2) * BesselK(nu, sqrt(j) * x) where j = sqrt(-1). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> nu</h6>
|
|
<p class="comments">the order of the the Kelvin function. </p>
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The non-negative real value to compute the Kelvin function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The Kelvin function kei. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="KelvinKeiPrime" class="method">
|
|
<h4><span title="System.double">double</span> <strong>KelvinKeiPrime</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the derivative of the Kelvin function kei.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the derivative of the Kelvin function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The derivative of the Kelvin function kei. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="KelvinKeiPrime" class="method">
|
|
<h4><span title="System.double">double</span> <strong>KelvinKeiPrime</strong>(<span title="System.double">double</span> nu, <span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the derivative of the Kelvin function kei.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> nu</h6>
|
|
<p class="comments">The order of the Kelvin function. </p>
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the derivative of the Kelvin function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The derivative of the Kelvin function kei. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="KelvinKer" class="method">
|
|
<h4><span title="System.double">double</span> <strong>KelvinKer</strong>(<span title="System.double">double</span> nu, <span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the Kelvin function ker. <p>KelvinKer(nu, x) is given by the real part of Exp(-nu * pi * j / 2) * BesselK(nu, sqrt(j) * x) where j = sqrt(-1). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> nu</h6>
|
|
<p class="comments">the order of the the Kelvin function. </p>
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The non-negative real value to compute the Kelvin function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The Kelvin function ker. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="KelvinKer" class="method">
|
|
<h4><span title="System.double">double</span> <strong>KelvinKer</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the Kelvin function ker. <p>KelvinKer(x) is given by the real part of Exp(-nu * pi * j / 2) * BesselK(0, sqrt(j) * x) where j = sqrt(-1). </p> <p>KelvinKer(x) is equivalent to KelvinKer(0, x). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The non-negative real value to compute the Kelvin function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The Kelvin function ker. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="KelvinKerPrime" class="method">
|
|
<h4><span title="System.double">double</span> <strong>KelvinKerPrime</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the derivative of the Kelvin function ker.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the derivative of the Kelvin function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The derivative of the Kelvin function ker. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="KelvinKerPrime" class="method">
|
|
<h4><span title="System.double">double</span> <strong>KelvinKerPrime</strong>(<span title="System.double">double</span> nu, <span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the derivative of the Kelvin function ker.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> nu</h6>
|
|
<p class="comments">The order of the Kelvin function. </p>
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The non-negative real value to compute the derivative of the Kelvin function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The derivative of the Kelvin function ker. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="Logistic" class="method">
|
|
<h4><span title="System.double">double</span> <strong>Logistic</strong>(<span title="System.double">double</span> p)</h4>
|
|
<div class="content">Computes the logistic function. see: http://en.wikipedia.org/wiki/Logistic
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> p</h6>
|
|
<p class="comments">The parameter for which to compute the logistic function. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The logistic function of <var>p</var>. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="Logit" class="method">
|
|
<h4><span title="System.double">double</span> <strong>Logit</strong>(<span title="System.double">double</span> p)</h4>
|
|
<div class="content">Computes the logit function, the inverse of the sigmoid logistic function. see: http://en.wikipedia.org/wiki/Logit
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> p</h6>
|
|
<p class="comments">The parameter for which to compute the logit function. This number should be
|
|
between 0 and 1. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The logarithm of <var>p</var> divided by 1.0 - <var>p</var>. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="MarcumQ" class="method">
|
|
<h4><span title="System.double">double</span> <strong>MarcumQ</strong>(<span title="System.double">double</span> nu, <span title="System.double">double</span> a, <span title="System.double">double</span> b)</h4>
|
|
<div class="content">Returns the Marcum Q-function Q[ν](a,b). <p>References: A. Gil, J. Segura and N.M. Temme. Efficient and accurate algorithms for the
|
|
computation and inversion of the incomplete gamma function ratios. SIAM J Sci Comput. (2012) 34(6), A2965-A2981 </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> nu</h6>
|
|
<p class="comments">The order of generalized Marcum Q-function. Range: 1≦ν≦10000 </p>
|
|
<h6><code><span title="System.double">double</span></code> a</h6>
|
|
<p class="comments">The value to compute the Marcum Q-function of. Range: 0≦a≦10000 </p>
|
|
<h6><code><span title="System.double">double</span></code> b</h6>
|
|
<p class="comments">The value to compute the Marcum Q-function of. Range: 0≦b≦10000 </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The Marcum Q-function Q[ν](a,b) </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="MarcumQ" class="method">
|
|
<h4><span title="System.double">double</span> <strong>MarcumQ</strong>(<span title="System.double">double</span> nu, <span title="System.double">double</span> a, <span title="System.double">double</span> b, <span title="System.Int32&">Int32&</span> err)</h4>
|
|
<div class="content">
|
|
|
|
|
|
|
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</div>
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</div>
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<div id="Multinomial" class="method">
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<h4><span title="System.double">double</span> <strong>Multinomial</strong>(<span title="System.int">int</span> n, <span title="System.Int32[]">Int32[]</span> ni)</h4>
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<div class="content">Computes the multinomial coefficient: n choose n1, n2, n3,...
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<div class="parameters">
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<h5>Parameters</h5>
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<h6><code><span title="System.int">int</span></code> n</h6>
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<p class="comments">A nonnegative value n. </p>
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<h6><code><span title="System.Int32[]">Int32[]</span></code> ni</h6>
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<p class="comments">An array of nonnegative values that sum to <var>n</var>. </p>
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</div>
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<div class="return">
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<h5>Return</h5>
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<h6><code><span title="System.double">double</span></code></h6>
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<p>The multinomial coefficient. </p>
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</div>
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</div>
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</div>
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<div id="RisingFactorial" class="method">
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<h4><span title="System.double">double</span> <strong>RisingFactorial</strong>(<span title="System.double">double</span> x, <span title="System.int">int</span> n)</h4>
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<div class="content">Computes the Rising Factorial (Pochhammer function) x -> (x)n, n>= 0. see: https://en.wikipedia.org/wiki/Falling_and_rising_factorials
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<div class="return">
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<h5>Return</h5>
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<h6><code><span title="System.double">double</span></code></h6>
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<p>The real value of the Rising Factorial for x and n </p>
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</div>
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</div>
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</div>
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<div id="SphericalBesselJ" class="method">
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|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>SphericalBesselJ</strong>(<span title="System.double">double</span> n, <span title="System.Numerics.Complex">Complex</span> z)</h4>
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<div class="content">Returns the spherical Bessel function of the first kind. <p>SphericalBesselJ(n, z) is given by Sqrt(pi/2) / Sqrt(z) * BesselJ(n + 1/2, z). </p>
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<div class="parameters">
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<h5>Parameters</h5>
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<h6><code><span title="System.double">double</span></code> n</h6>
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<p class="comments">The order of the spherical Bessel function. </p>
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<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
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<p class="comments">The value to compute the spherical Bessel function of. </p>
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</div>
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<div class="return">
|
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<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The spherical Bessel function of the first kind. </p>
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</div>
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</div>
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</div>
|
|
<div id="SphericalBesselJ" class="method">
|
|
<h4><span title="System.double">double</span> <strong>SphericalBesselJ</strong>(<span title="System.double">double</span> n, <span title="System.double">double</span> z)</h4>
|
|
<div class="content">Returns the spherical Bessel function of the first kind. <p>SphericalBesselJ(n, z) is given by Sqrt(pi/2) / Sqrt(z) * BesselJ(n + 1/2, z). </p>
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<div class="parameters">
|
|
<h5>Parameters</h5>
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|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the spherical Bessel function. </p>
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The value to compute the spherical Bessel function of. </p>
|
|
</div>
|
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|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The spherical Bessel function of the first kind. </p>
|
|
</div>
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|
</div>
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</div>
|
|
<div id="SphericalBesselY" class="method">
|
|
<h4><span title="System.Numerics.Complex">Complex</span> <strong>SphericalBesselY</strong>(<span title="System.double">double</span> n, <span title="System.Numerics.Complex">Complex</span> z)</h4>
|
|
<div class="content">Returns the spherical Bessel function of the second kind. <p>SphericalBesselY(n, z) is given by Sqrt(pi/2) / Sqrt(z) * BesselY(n + 1/2, z). </p>
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|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the spherical Bessel function. </p>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code> z</h6>
|
|
<p class="comments">The value to compute the spherical Bessel function of. </p>
|
|
</div>
|
|
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|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.Numerics.Complex">Complex</span></code></h6>
|
|
<p>The spherical Bessel function of the second kind. </p>
|
|
</div>
|
|
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|
</div>
|
|
</div>
|
|
<div id="SphericalBesselY" class="method">
|
|
<h4><span title="System.double">double</span> <strong>SphericalBesselY</strong>(<span title="System.double">double</span> n, <span title="System.double">double</span> z)</h4>
|
|
<div class="content">Returns the spherical Bessel function of the second kind. <p>SphericalBesselY(n, z) is given by Sqrt(pi/2) / Sqrt(z) * BesselY(n + 1/2, z). </p>
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> n</h6>
|
|
<p class="comments">The order of the spherical Bessel function. </p>
|
|
<h6><code><span title="System.double">double</span></code> z</h6>
|
|
<p class="comments">The value to compute the spherical Bessel function of. </p>
|
|
</div>
|
|
|
|
<div class="return">
|
|
<h5>Return</h5>
|
|
<h6><code><span title="System.double">double</span></code></h6>
|
|
<p>The spherical Bessel function of the second kind. </p>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<div id="StruveL0" class="method">
|
|
<h4><span title="System.double">double</span> <strong>StruveL0</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the modified Struve function of order 0.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the function of. </p>
|
|
</div>
|
|
|
|
|
|
</div>
|
|
</div>
|
|
<div id="StruveL1" class="method">
|
|
<h4><span title="System.double">double</span> <strong>StruveL1</strong>(<span title="System.double">double</span> x)</h4>
|
|
<div class="content">Returns the modified Struve function of order 1.
|
|
|
|
|
|
<div class="parameters">
|
|
<h5>Parameters</h5>
|
|
|
|
<h6><code><span title="System.double">double</span></code> x</h6>
|
|
<p class="comments">The value to compute the function of. </p>
|
|
</div>
|
|
|
|
|
|
</div>
|
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</div>
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<div id="footer">
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<p>Based on v5.0.0.0 of MathNet.Numerics (Math.NET Numerics)</p>
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<p>Generated by <a href="http://docu.jagregory.com">docu</a></p>
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