Formula:KLS:01.12:03

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${\displaystyle{\displaystyle{\displaystyle\int_{0}^{\infty}\frac{\left(-ax,-bq/x;q\right)_{\infty}}{\left(-x,-q/x;q\right)_{\infty}}\,dx=-\ln q\,\frac{\left(ab,q;q\right)_{\infty}}{\left(bq,a/q;q\right)_{\infty}}}}}$

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
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2

Bibliography

Equation in Section 1.12 of KLS.

URL links

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