DLMF:13.11.E1

MaRDI QIDQ5173

• Label: DLMF:13.11.E1

Digital Library of Mathematical Functions ID 13.11.E1

${\displaystyle{\displaystyle M\left(a,b,z\right)=\Gamma\left(a-\tfrac{1}{2}\right)e^{\frac{1}{2}z}\left(\tfrac{1}{4}z\right)^{\frac{1}{2}-a}\*\sum_{s=0}^{\infty}\frac{{\left(2a-1\right)_{s}}{\left(2a-b\right)_{s}}}{{\left(b\right)_{s}}s!}\*\left(a-\tfrac{1}{2}+s\right)\*I_{a-\frac{1}{2}+s}\left(\tfrac{1}{2}z\right),}}$

Constraint(s)

${\displaystyle{\displaystyle a+\frac{1}{2},b\neq 0,-1,-2,\dots}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle M\left(\NVar{a},\NVar{b},\NVar{z}\right)}}$: $$={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$$ Kummer confluent hypergeometric function
• ${\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}$:
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}$: modified Bessel function of the first kind
• ${\displaystyle{\displaystyle s}}$: nonnegative integer
• ${\displaystyle{\displaystyle z}}$: complex variable
 

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