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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)
 > $\frac{\partial }{\partial z}\mathrm{WhittakerW}\left(\mathrm{μ},\mathrm{ν},z\right)$
 $\left(\frac{{1}}{{2}}{-}\frac{{\mathrm{\mu }}}{{z}}\right){}{\mathrm{WhittakerW}}{}\left({\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{z}\right){-}\frac{{\mathrm{WhittakerW}}{}\left({\mathrm{\mu }}{+}{1}{,}{\mathrm{\nu }}{,}{z}\right)}{{z}}$ (2)
 > $\mathrm{series}\left(\mathrm{WhittakerM}\left(2,3,x\right),x\right)$
 ${{x}}^{{7}}{{2}}}{-}\frac{{2}{}{{x}}^{{9}}{{2}}}}{{7}}{+}\frac{{23}{}{{x}}^{{11}}{{2}}}}{{448}}{+}{\mathrm{O}}{}\left({{x}}^{{13}}{{2}}}\right)$ (3)
 > $\mathrm{series}\left(\mathrm{WhittakerW}\left(-\frac{1}{2},-\frac{1}{3},x\right),x\right)$
 $\frac{{3}{}\sqrt{{3}}{}{{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}^{{2}}{}{{x}}^{{1}}{{6}}}}{{2}{}{\mathrm{\pi }}}{-}\frac{{\mathrm{\pi }}{}\sqrt{{3}}{}{{x}}^{{5}}{{6}}}}{{{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}^{{2}}}{+}\frac{{9}{}\sqrt{{3}}{}{{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}^{{2}}{}{{x}}^{{7}}{{6}}}}{{4}{}{\mathrm{\pi }}}{-}\frac{{3}{}{\mathrm{\pi }}{}\sqrt{{3}}{}{{x}}^{{11}}{{6}}}}{{10}{}{{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}^{{2}}}{+}\frac{{9}{}\sqrt{{3}}{}{{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}^{{2}}{}{{x}}^{{13}}{{6}}}}{{16}{}{\mathrm{\pi }}}{-}\frac{{3}{}{\mathrm{\pi }}{}\sqrt{{3}}{}{{x}}^{{17}}{{6}}}}{{40}{}{{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}^{{2}}}{+}\frac{{27}{}\sqrt{{3}}{}{{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}^{{2}}{}{{x}}^{{19}}{{6}}}}{{224}{}{\mathrm{\pi }}}{-}\frac{{9}{}{\mathrm{\pi }}{}\sqrt{{3}}{}{{x}}^{{23}}{{6}}}}{{880}{}{{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}^{{2}}}{+}\frac{{27}{}\sqrt{{3}}{}{{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}^{{2}}{}{{x}}^{{25}}{{6}}}}{{1792}{}{\mathrm{\pi }}}{-}\frac{{9}{}{\mathrm{\pi }}{}\sqrt{{3}}{}{{x}}^{{29}}{{6}}}}{{7040}{}{{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}^{{2}}}{+}\frac{{81}{}\sqrt{{3}}{}{{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}^{{2}}{}{{x}}^{{31}}{{6}}}}{{46592}{}{\mathrm{\pi }}}{-}\frac{{27}{}{\mathrm{\pi }}{}\sqrt{{3}}{}{{x}}^{{35}}{{6}}}}{{239360}{}{{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}^{{2}}}{+}{\mathrm{O}}{}\left({{x}}^{{37}}{{6}}}\right)$ (4)
 > $\mathrm{simplify}\left(\mathrm{WhittakerW}\left(\mathrm{μ}+\frac{7}{3},\mathrm{ν},x\right)\right)$
 ${-}\left({\mathrm{\nu }}{+}{\mathrm{\mu }}{-}\frac{{1}}{{6}}\right){}\left({x}{-}{2}{}{\mathrm{\mu }}{-}\frac{{8}}{{3}}\right){}\left({\mathrm{\mu }}{-}\frac{{1}}{{6}}{-}{\mathrm{\nu }}\right){}{\mathrm{WhittakerW}}{}\left({\mathrm{\mu }}{-}\frac{{2}}{{3}}{,}{\mathrm{\nu }}{,}{x}\right){+}\left({5}{}{{\mathrm{\mu }}}^{{2}}{+}\left({-}{4}{}{x}{+}\frac{{25}}{{3}}\right){}{\mathrm{\mu }}{+}{{x}}^{{2}}{-}{{\mathrm{\nu }}}^{{2}}{-}\frac{{10}{}{x}}{{3}}{+}\frac{{89}}{{36}}\right){}{\mathrm{WhittakerW}}{}\left({\mathrm{\mu }}{+}\frac{{1}}{{3}}{,}{\mathrm{\nu }}{,}{x}\right)$ (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.