Sunday, 28 May 2017

integration - show that intpi/20tanax,dx=fracpi2cos(fracpia2)




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



keyhole



we find that



(1ei2πa)0duua1+u2=i2πeiπa/2ei3aπ/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+iR2π0dθeiθRaeiaθ1+R2ei2θ+ei2πaϵRdxxa1+x2+iϵ2π0dϕeiϕϵaeiaϕ1+ϵ2ei2ϕ



We take the limit as R and ϵ0 and we recover the expression for the contour integral above.


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