WhittakerM - Maple Help

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WhittakerM

The Whittaker M function

WhittakerW

The Whittaker W function

 Calling Sequence WhittakerM(mu, nu, z) WhittakerW(mu, nu, z)

Parameters

 mu - algebraic expression nu - algebraic expression z - algebraic expression

Description

 • The Whittaker functions WhittakerM(mu, nu, z) and WhittakerW(mu, nu, z) solve the differential equation

$\mathrm{y\text{'}\text{'}}+\left(-\frac{1}{4}+\frac{\mathrm{\mu }}{z}+\frac{\frac{1}{4}-{\mathrm{\nu }}^{2}}{{z}^{2}}\right)y=0$

 • They can be defined in terms of the hypergeometric and Kummer functions as follows:

$\mathrm{WhittakerM}\left(\mathrm{\mu },\mathrm{\nu },z\right)={ⅇ}^{-\frac{1}{2}z}{z}^{\frac{1}{2}+\mathrm{\nu }}\mathrm{hypergeom}\left(\left[\frac{1}{2}+\mathrm{\nu }-\mathrm{\mu }\right],\left[1+2\mathrm{\nu }\right],z\right)$

$\mathrm{WhittakerW}\left(\mathrm{\mu },\mathrm{\nu },z\right)={ⅇ}^{-\frac{1}{2}z}{z}^{\frac{1}{2}+\mathrm{\nu }}\mathrm{KummerU}\left(\frac{1}{2}+\mathrm{\nu }-\mathrm{\mu },1+2\mathrm{\nu },z\right)$

Examples

 > $\mathrm{WhittakerM}\left(1,2,0.5\right)$
 ${0.1606687379}$ (1)
 > $\frac{\partial }{\partial z}\mathrm{WhittakerW}\left(\mathrm{μ},\mathrm{ν},z\right)$
 $\left(\frac{{1}}{{2}}{-}\frac{{\mathrm{μ}}}{{z}}\right){}{\mathrm{WhittakerW}}{}\left({\mathrm{μ}}{,}{\mathrm{ν}}{,}{z}\right){-}\frac{{\mathrm{WhittakerW}}{}\left({\mathrm{μ}}{+}{1}{,}{\mathrm{ν}}{,}{z}\right)}{{z}}$ (2)
 > $\mathrm{series}\left(\mathrm{WhittakerM}\left(2,3,x\right),x\right)$
 ${{x}}^{{7}{/}{2}}{-}\frac{{2}}{{7}}{}{{x}}^{{9}{/}{2}}{+}\frac{{23}}{{448}}{}{{x}}^{{11}{/}{2}}{+}{\mathrm{O}}{}\left({{x}}^{{13}{/}{2}}\right)$ (3)
 > $\mathrm{series}\left(\mathrm{WhittakerW}\left(-\frac{1}{2},-\frac{1}{3},x\right),x\right)$
 $\frac{{3}}{{2}}{}\frac{\sqrt{{3}}{}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}{}{{x}}^{{1}{/}{6}}}{{\mathrm{π}}}{-}\frac{{\mathrm{π}}{}\sqrt{{3}}{}{{x}}^{{5}{/}{6}}}{{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}}{+}\frac{{9}}{{4}}{}\frac{\sqrt{{3}}{}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}{}{{x}}^{{7}{/}{6}}}{{\mathrm{π}}}{-}\frac{{3}}{{10}}{}\frac{{\mathrm{π}}{}\sqrt{{3}}{}{{x}}^{{11}{/}{6}}}{{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}}{+}\frac{{9}}{{16}}{}\frac{\sqrt{{3}}{}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}{}{{x}}^{{13}{/}{6}}}{{\mathrm{π}}}{-}\frac{{3}}{{40}}{}\frac{{\mathrm{π}}{}\sqrt{{3}}{}{{x}}^{{17}{/}{6}}}{{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}}{+}\frac{{27}}{{224}}{}\frac{\sqrt{{3}}{}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}{}{{x}}^{{19}{/}{6}}}{{\mathrm{π}}}{-}\frac{{9}}{{880}}{}\frac{{\mathrm{π}}{}\sqrt{{3}}{}{{x}}^{{23}{/}{6}}}{{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}}{+}\frac{{27}}{{1792}}{}\frac{\sqrt{{3}}{}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}{}{{x}}^{{25}{/}{6}}}{{\mathrm{π}}}{-}\frac{{9}}{{7040}}{}\frac{{\mathrm{π}}{}\sqrt{{3}}{}{{x}}^{{29}{/}{6}}}{{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}}{+}\frac{{81}}{{46592}}{}\frac{\sqrt{{3}}{}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}{}{{x}}^{{31}{/}{6}}}{{\mathrm{π}}}{-}\frac{{27}}{{239360}}{}\frac{{\mathrm{π}}{}\sqrt{{3}}{}{{x}}^{{35}{/}{6}}}{{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}}{+}{\mathrm{O}}{}\left({{x}}^{{37}{/}{6}}\right)$ (4)
 > $\mathrm{simplify}\left(\mathrm{WhittakerW}\left(\mathrm{μ}+\frac{7}{3},\mathrm{ν},x\right)\right)$
 $\left({2}{}{\mathrm{μ}}{+}\frac{{8}}{{3}}{-}{x}\right){}\left(\left({2}{}{\mathrm{μ}}{+}\frac{{2}}{{3}}{-}{x}\right){}{\mathrm{WhittakerW}}{}\left({\mathrm{μ}}{+}\frac{{1}}{{3}}{,}{\mathrm{ν}}{,}{x}\right){-}\left(\frac{{1}}{{6}}{-}{\mathrm{μ}}{+}{\mathrm{ν}}\right){}\left({\mathrm{ν}}{+}{\mathrm{μ}}{-}\frac{{1}}{{6}}\right){}{\mathrm{WhittakerW}}{}\left({\mathrm{μ}}{-}\frac{{2}}{{3}}{,}{\mathrm{ν}}{,}{x}\right)\right){-}\left({-}\frac{{5}}{{6}}{-}{\mathrm{μ}}{+}{\mathrm{ν}}\right){}\left({\mathrm{ν}}{+}{\mathrm{μ}}{+}\frac{{5}}{{6}}\right){}{\mathrm{WhittakerW}}{}\left({\mathrm{μ}}{+}\frac{{1}}{{3}}{,}{\mathrm{ν}}{,}{x}\right)$ (5)

References

 Abramowitz, M., and Stegun I. Handbook of Mathematical Functions. New York: Dover Publications.
 Luke, Y. The Special Functions and Their Approximations. Vol 1. Academic Press, 1969.