DLMF:8.14.E6 (Q2639)

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Language	Label	Description	Also known as
English	DLMF:8.14.E6	No description defined

Statements

Defining formula

{\displaystyle{\displaystyle\int_{0}^{\infty}x^{a-1}e^{-sx}\Gamma\left(b,x% \right)\mathrm{d}x=\frac{\Gamma\left(a+b\right)}{a(1+s)^{a+b}}\*F\left(1,a+b;1% +a;s/(1+s)\right),}}

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DLMF:8.14.E6

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{\displaystyle{\displaystyle\Re s>-1}}

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{\displaystyle{\displaystyle\Re(a+b)>0}}

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{\displaystyle{\displaystyle\Re a>0}}

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Defining formula

{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

C5.S2.E1.m2aedec

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$$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function

Defining formula

{\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}

C15.S2.E1.m2aadec

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Defining formula

{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

C1.S4.SS4.m1aedec

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base of natural logarithm

Defining formula

{\displaystyle{\displaystyle\mathrm{e}}}

C4.S2.E11.m2acdec

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incomplete gamma function

Defining formula

{\displaystyle{\displaystyle\Gamma\left(\NVar{a},\NVar{z}\right)}}

C8.S2.E2.m2abdec

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Defining formula

{\displaystyle{\displaystyle\int}}

C1.S4.SS4.m3aedec

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Defining formula

{\displaystyle{\displaystyle\Re}}

C1.S9.E2.m1aedec

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