show that ∫π/20tanaxdx=π2cos(πa2)
I think we can solve it by contour integration but I dont know how.
If someone can solve it by two way using complex and real analysis its better for me.
thanks for all.
Answer
Let u=tanx, dx=du/(1+u2). Then the integral is
∫∞0duua1+u2
This integral may be performed for a∈(−1,1) by residue theory. By considering a contour integral about a keyhole contour about the positive real axis
we find that
(1−ei2πa)∫∞0duua1+u2=i2πeiπa/2−ei3aπ/22i
Or
∫∞0duua1+u2=πsinπa/2sinπa
From which the sought after result may be found.
ADDENDUM
A little further explanation. Consider the contour integral
∮Cdzza1+z2
where C is the above keyhole contour. This means that the integral may be written as
∫Rϵdxxa1+x2+iR∫2π0dθeiθRaeiaθ1+R2ei2θ+ei2πa∫ϵRdxxa1+x2+iϵ∫2π0dϕeiϕϵaeiaϕ1+ϵ2ei2ϕ
We take the limit as R→∞ and ϵ→0 and we recover the expression for the contour integral above.
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