Formula:KLS:01.12:04

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Formula:KLS:01.13:01 >>

${\displaystyle{\displaystyle{\displaystyle{}{}{}\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x)}{\sqrt{1-x^{2}}}\,dx=\frac{1}{2\pi}\int_{0}^{\pi}w(\cos\theta)\,d\theta=\frac{\left(abcd;q\right)_{\infty}}{\left(ab,ac,ad,bc,bd,cd,q;q\right)_{\infty}}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle w(x)=\frac{h(x,1)h(x,-1)h(x,q^{1/2})h(x,-q^{1/2})}{h(x,a)h(x,b)h(x,c)h(x,d)}=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}},c{\mathrm{e}^{\mathrm{i}\theta}},d{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}},{\mathrm{e}^{-2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm{i}\theta}},a{\mathrm{e}^{-\mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}},b{\mathrm{e}^{-\mathrm{i}\theta}},c{\mathrm{e}^{\mathrm{i}\theta}},c{\mathrm{e}^{-\mathrm{i}\theta}},d{\mathrm{e}^{\mathrm{i}\theta}},d{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}}}$ &

${\displaystyle{\displaystyle{\displaystyle h(x,\alpha)=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left|\left(\alpha{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}\right|^{2}=\left(\alpha{\mathrm{e}^{\mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}}$ &

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\Pi}}}$ : product : http://drmf.wmflabs.org/wiki/Definition:prod

Bibliography

Equation in Section 1.12 of KLS.

URL links

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