Sunday, 9 March 2014

complex analysis - Residue integral: int+inftyinftyfraceax1+exdx with 0ltalt1.



I'm self studying complex analysis. I've encountered the following integral:



+eax1+exdx with aR, 0<a<1.




I've done the substitution ex=y. What kind of contour can I use in this case ?


Answer



The substitution ex=y leads to the integral
0ya1y+1dy.


This can be computed integrating the function f(z)=za1/(z+1) along the keyhole contour. We consider the branch of za1 defined on C[0,) with f(1)=eπi. For small ϵ>0 and large R>0, he contour is made up of the interval [ϵ,R], the circle Cr={|z|=R} counterclockwise, the interval [R,ϵ] and the circle Cϵ={|z|=ϵ} clockwise. The function f has a simple pole at z=1 with residue (1)a1=eπ(a1)i. It is easy to see that
limϵ0Cϵf(z)dz=limRCRf(z)dz=0.


Then
0ya1y+1dy+0(e2πiy)a1y+1dy=2πiRes(f,1),

from where
(1e2π(a1)i)0ya1y+1dy=2πieπ(a1)i

and
0ya1y+1dy=πsin((1a)π).


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