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turtlejohn January 20th, 2010 08:49 PM

how to support this step mathematically
 
Here is a problem which uses the residue theorem to evaluate a real-valued 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 semi-circle as a contour in the upper-half 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 upper-half 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 (z-i) 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 (z-i) from the natural logarithm in the numerator of f(z)?

Please help. Thanks.


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