Formula:KLS:01.12:02

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Formula:KLS:01.12:03 >>

${\displaystyle{\displaystyle{\displaystyle\int_{0}^{\infty}x^{c-1}\frac{\left(-ax;q\right)_{\infty}}{\left(-x;q\right)_{\infty}}\,dx=\frac{\pi}{\sin\left(\pi c\right)}\,\frac{\left(a,q^{1-c};q\right)_{\infty}}{\left(aq^{-c},q;q\right)_{\infty}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}$ : sine function : http://dlmf.nist.gov/4.14#E1

Bibliography

Equation in Section 1.12 of KLS.

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