Question Number 221882 by MrGaster last updated on 12/Jun/25
$$\mathrm{Prove}:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{sin}\:{x}\:{K}^{\mathrm{2}} \:\mathrm{sin}\:{x}\:{dx}=\frac{\pi^{\mathrm{4}} }{\mathrm{16}}\:_{\mathrm{7}} {F}_{\mathrm{6}} \left(\frac{\mathrm{1}}{\:\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{5}}{\mathrm{4}};\mathrm{1},\mathrm{1},\mathrm{1},\mathrm{1},\mathrm{1},\frac{\mathrm{1}}{\mathrm{4}};\mathrm{1}\right) \\ $$
Answered by wewji12 last updated on 12/Jun/25
$${K}^{\mathrm{2}} ….?? \\ $$$$\mathrm{you}\:\mathrm{mean}\:{K}\left({k}\right)=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}−{k}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right)}}\:\mathrm{d}\theta..?? \\ $$$$\mathrm{a}.\mathrm{k}.\mathrm{a}\:\mathrm{Eliptic}\:\mathrm{integration}\:\mathrm{function}\:\mathrm{fisrt}\:\mathrm{kind} \\ $$
Commented by MrGaster last updated on 12/Jun/25
yes(Hypergeometry and 4-fold Pochhammer symbols are required)