DLMF:15.6.E2 (Q5040)

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Language	Label	Description	Also known as
English	DLMF:15.6.E2	No description defined

Statements

Defining formula

{\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\frac{\Gamma\left(1% +b-c\right)}{2\pi\mathrm{i}\Gamma\left(b\right)}\int_{0}^{(1+)}\frac{t^{b-1}(t% -1)^{c-b-1}}{(1-zt)^{a}}\mathrm{d}t,}}

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DLMF:15.6.E2

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{\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}

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{\displaystyle{\displaystyle\Re b>0}}

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{\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}

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{\displaystyle{\displaystyle c-b\neq 1,2,3,\dots}}

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{\displaystyle{\displaystyle\Re b>0}}

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Defining formula

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

C5.S2.E1.m2aadec

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the ratio of the circumference of a circle to its diameter

Defining formula

{\displaystyle{\displaystyle\pi}}

C3.S12.E1.m2aadec

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Defining formula

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

C1.S4.SS4.m1aadec

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Defining formula

{\displaystyle{\displaystyle\mathrm{i}}}

C1.S9.E1.m2adec

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Defining formula

{\displaystyle{\displaystyle\int}}

C1.S4.SS4.m3aadec

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Defining formula

{\displaystyle{\displaystyle\operatorname{ph}}}

C1.S9.E7.m1aadec

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Defining formula

{\displaystyle{\displaystyle\Re}}

C1.S9.E2.m1aadec

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$$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function

Defining formula

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

C15.S2.E2.m2aadec

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Defining formula

{\displaystyle{\displaystyle z}}

C15.S1.XMD3.m1adec

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Defining formula

{\displaystyle{\displaystyle a}}

C15.S1.XMD4.m1adec

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Defining formula

{\displaystyle{\displaystyle b}}

C15.S1.XMD5.m1adec

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Defining formula

{\displaystyle{\displaystyle c}}

C15.S1.XMD6.m1adec

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