KummerM - Maple Programming Help

KummerM

The Kummer M_{mu,nu}(z) function

KummerU

The Kummer U_{mu,nu}(z) function

 Calling Sequence KummerM(mu, nu, z) KummerU(mu, nu, z)

Parameters

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

Description

 • The Kummer functions KummerM(mu, nu, z) and KummerU(mu, nu, z) solve the differential equation

$zy\text{'}\text{'}+\left(\mathrm{nu}-z\right)y'-\mathrm{mu}y=0$

Examples

 > $\mathrm{KummerM}\left(1,2,0.5\right)$
 ${1.297442541}$ (1)
 > $\mathrm{evalf}\left(\mathrm{KummerU}\left(-\frac{1}{2},-\frac{1}{3},\frac{1}{7}\right)\right)$
 ${0.9025082951}$ (2)
 > $\frac{\partial }{\partial z}\mathrm{KummerM}\left(\mathrm{μ},\mathrm{ν},z\right)$
 $\frac{\left({z}{+}{\mathrm{μ}}{-}{\mathrm{ν}}\right){}{\mathrm{KummerM}}{}\left({\mathrm{μ}}{,}{\mathrm{ν}}{,}{z}\right){+}\left({\mathrm{ν}}{-}{\mathrm{μ}}\right){}{\mathrm{KummerM}}{}\left({-}{1}{+}{\mathrm{μ}}{,}{\mathrm{ν}}{,}{z}\right)}{{z}}$ (3)
 > $\mathrm{series}\left(\mathrm{KummerU}\left(\frac{1}{2},\frac{1}{3},z\right),z\right)$
 $\frac{{3}{}{\mathrm{Γ}}{}\left(\frac{{5}}{{6}}\right){}{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}{{\mathrm{π}}}{-}\frac{\sqrt{{\mathrm{π}}}{}\sqrt{{3}}{}{{z}}^{{2}{/}{3}}}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}{+}\frac{{9}}{{2}}{}\frac{{\mathrm{Γ}}{}\left(\frac{{5}}{{6}}\right){}{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right){}{z}}{{\mathrm{π}}}{-}\frac{{7}}{{10}}{}\frac{\sqrt{{\mathrm{π}}}{}\sqrt{{3}}{}{{z}}^{{5}{/}{3}}}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}{+}\frac{{81}}{{32}}{}\frac{{\mathrm{Γ}}{}\left(\frac{{5}}{{6}}\right){}{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right){}{{z}}^{{2}}}{{\mathrm{π}}}{-}\frac{{91}}{{320}}{}\frac{\sqrt{{\mathrm{π}}}{}\sqrt{{3}}{}{{z}}^{{8}{/}{3}}}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}{+}\frac{{405}}{{448}}{}\frac{{\mathrm{Γ}}{}\left(\frac{{5}}{{6}}\right){}{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right){}{{z}}^{{3}}}{{\mathrm{π}}}{-}\frac{{1729}}{{21120}}{}\frac{\sqrt{{\mathrm{π}}}{}\sqrt{{3}}{}{{z}}^{{11}{/}{3}}}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}{+}\frac{{243}}{{1024}}{}\frac{{\mathrm{Γ}}{}\left(\frac{{5}}{{6}}\right){}{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right){}{{z}}^{{4}}}{{\mathrm{π}}}{-}\frac{{1235}}{{67584}}{}\frac{\sqrt{{\mathrm{π}}}{}\sqrt{{3}}{}{{z}}^{{14}{/}{3}}}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}{+}\frac{{6561}}{{133120}}{}\frac{{\mathrm{Γ}}{}\left(\frac{{5}}{{6}}\right){}{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right){}{{z}}^{{5}}}{{\mathrm{π}}}{-}\frac{{7657}}{{2297856}}{}\frac{\sqrt{{\mathrm{π}}}{}\sqrt{{3}}{}{{z}}^{{17}{/}{3}}}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}{+}{\mathrm{O}}{}\left({{z}}^{{6}}\right)$ (4)

References

 Abramowitz, M., and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover, 1972.