View source for 10.17

{{DISPLAYTITLE:Bessel Functions - 10.17 Asymptotic Expansions for Large Argument}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/10.17.E7 10.17.E7] || <math qid="Q3176">z^{\frac{1}{2}} = \exp@{\tfrac{1}{2}\ln@@{|z|}+\tfrac{1}{2}i\phase@@{z}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z^{\frac{1}{2}} = \exp@{\tfrac{1}{2}\ln@@{|z|}+\tfrac{1}{2}i\phase@@{z}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(z)^((1)/(2)) = exp((1)/(2)*ln(abs(z))+(1)/(2)*I*argument(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(z)^(Divide[1,2]) == Exp[Divide[1,2]*Log[Abs[z]]+Divide[1,2]*I*Arg[z]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 7] || Successful [Tested: 7]
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| [https://dlmf.nist.gov/10.17.E16 10.17.E16] || <math qid="Q3185">\scterminant{p}@{z} = \frac{e^{z}}{2\pi}\EulerGamma@{p}\incGamma@{1-p}{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\scterminant{p}@{z} = \frac{e^{z}}{2\pi}\EulerGamma@{p}\incGamma@{1-p}{z}</syntaxhighlight> || <math>\realpart@@{p} > 0</math> || <syntaxhighlight lang=mathematica>(exp(z)/(2*Pi))*GAMMA(p)*GAMMA(1-p,z) = (exp(z))/(2*Pi)*GAMMA(p)*GAMMA(1 - p, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Successful || Missing Macro Error || - || -
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| [https://dlmf.nist.gov/10.17.E17 10.17.E17] || <math qid="Q3186">R_{\ell}^{+}(\nu,z) = (-1)^{\ell}2\cos@{\nu\pi}\*\left(\sum_{k=0}^{m-1}(+ i)^{k}\frac{a_{k}(\nu)}{z^{k}}\scterminant{\ell-k}@{- 2iz}+R_{m,\ell}^{+}(\nu,z)\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>R_{\ell}^{+}(\nu,z) = (-1)^{\ell}2\cos@{\nu\pi}\*\left(\sum_{k=0}^{m-1}(+ i)^{k}\frac{a_{k}(\nu)}{z^{k}}\scterminant{\ell-k}@{- 2iz}+R_{m,\ell}^{+}(\nu,z)\right)</syntaxhighlight> || <math>\realpart@@{(\ell-k)} > 0, k \geq 1</math> || <syntaxhighlight lang=mathematica>(R[ell])^(+)(nu , z) = (- 1)^(ell)* 2*cos(nu*Pi)*(sum((+ I)^(k)*(((4*(nu)^(2)- (1)^(2))*(4*(nu)^(2)- (3)^(2)) .. (4*(nu)^(2)-(2*k - 1)^(2)))/(factorial(k)*(8)^(k)))/((z)^(k))*(exp(- 2*I*z)/(2*Pi))*GAMMA(ell - k)*GAMMA(1-ell - k,- 2*I*z), k = 0..m - 1)+ (R[m , ell])^(+)(nu , z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Error || Missing Macro Error || - || -
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| [https://dlmf.nist.gov/10.17.E17 10.17.E17] || <math qid="Q3186">R_{\ell}^{-}(\nu,z) = (-1)^{\ell}2\cos@{\nu\pi}\*\left(\sum_{k=0}^{m-1}(- i)^{k}\frac{a_{k}(\nu)}{z^{k}}\scterminant{\ell-k}@{+ 2iz}+R_{m,\ell}^{-}(\nu,z)\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>R_{\ell}^{-}(\nu,z) = (-1)^{\ell}2\cos@{\nu\pi}\*\left(\sum_{k=0}^{m-1}(- i)^{k}\frac{a_{k}(\nu)}{z^{k}}\scterminant{\ell-k}@{+ 2iz}+R_{m,\ell}^{-}(\nu,z)\right)</syntaxhighlight> || <math>\realpart@@{(\ell-k)} > 0, k \geq 1</math> || <syntaxhighlight lang=mathematica>(R[ell])^(-)(nu , z) = (- 1)^(ell)* 2*cos(nu*Pi)*(sum((- I)^(k)*(((4*(nu)^(2)- (1)^(2))*(4*(nu)^(2)- (3)^(2)) .. (4*(nu)^(2)-(2*k - 1)^(2)))/(factorial(k)*(8)^(k)))/((z)^(k))*(exp(+ 2*I*z)/(2*Pi))*GAMMA(ell - k)*GAMMA(1-ell - k,+ 2*I*z), k = 0..m - 1)+ (R[m , ell])^(-)(nu , z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Error || Missing Macro Error || - || -
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