Formula:KLS:01.12:01

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

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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<< The q-binomial theorem and other summation formulas

formula in More integrals

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