Question

How to use gamma function in integration trigonometric?

Answer 1

∫π20(sinx)z dx.∫0π2(sin⁡x)z dx.

I put this integral in Wolfram Alpha, and the result is

π−−√Γ(z+12)2Γ(z2+1),πΓ(z+12)2Γ(z2+1),

but I don't know why. If zz is a positive integer, then one can do integration by parts, many times. Eventually, this yields

∫π20(sinx)2z dx=(2z−1)!!(2z)!!π2,∫0π2(sin⁡x)2z dx=(2z−1)!!(2z)!!π2,

where (2n−1)!!=1⋅3⋯(2n−1)(2n−1)!!=1⋅3⋯(2n−1), and (2n)!!=2⋅4⋯2n(2n)!!=2⋅4⋯2n. I appreciate your help.

Answer 2

alpha plus beta plus gamma is equal to -B by A......