Formula:5030

Digital Library of Mathematical Functions ID 13.2.E41

${\displaystyle{\displaystyle\frac{1}{\Gamma\left(b\right)}M\left(a,b,z\right)=\frac{e^{\mp a\pi\mathrm{i}}}{\Gamma\left(b-a\right)}U\left(a,b,z\right)+\frac{e^{\pm(b-a)\pi\mathrm{i}}}{\Gamma\left(a\right)}e^{z}U\left(b-a,b,e^{\pm\pi\mathrm{i}}z\right).}}$

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
• ${\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}$: Kummer confluent hypergeometric function
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle z}}$: complex variable
 

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