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

References

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