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January 20th, 2010, 09:49 PM  #1 
Newbie Joined: Jan 2010 Posts: 1 Thanks: 0  how to support this step mathematically
Here is a problem which uses the residue theorem to evaluate a realvalued integral. evaluate the integral from 0 to infinity of In(x^2+1)/(x^2+1) dx I basically understand the solution to this integral given in the book. The book uses a semicircle as a contour in the upperhalf plane. But, the first step of the solution confuses me. This is the first step of the book's solution. step 1: let f(z)=In(z+i)/(z^2+1) I know that there is a simple pole of z=i in the upperhalf plane. If I let g(z)=In(z^2+1)/(z^2+1) instead, I realize that the function will diverge when you determine the residue at z=i since In(z^2+1) goes to infinity as z tends to i. I think this probably explains why the book removes the factor (zi) from the natural logarithm in the numerator of f(z) to prevent the function f(z) from blowing up at z=i. Is there a stronger argument to prove that we need to drop the factor (zi) from the natural logarithm in the numerator of f(z)? Please help. Thanks. 

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