Thursday, 9 June 2016

Evaluation of definite integral using complex analysis

I want to evaluate the following indefinite integral
0xp1cos(ax)dx where 0<p<1 and a>0. I was considering the function f(z)=zp1eiaz and integrate it over the contour γ=[R,ϵ]C+ϵ+[ϵ,R]+C+R. (Here C+r denote the upper half circle centered at 0 with radius r). However I only get
0xp1eiaxdxeiπp0xp1eiaxdx=0 instead, which only gives some relation between 0xp1sin(ax)dx and 0xp1cos(ax)dx. I know the result is somehow related to gamma function that is 0xp1cos(x)dx=π2Γ(1p)sin((1p)π/2) So would the regular method using complex analysis can still evaluate this integral?

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