In this turn, I will consider a couple of examples of complex contour integrals with respect to variable involving the following factor where a real parameter.
Problem 1. Evaluate the integral
and consider first the case . Then is bounded in the upper half-plane by
where is the semicircle centered at the origin joining and , oriented counterclockwise.
Then function is holomorphic on and its interior, so, by Cauchy’s Theorem, we have
The absolute value of the second summation on the right is at most
on . Taking the limit as we obtain .
Suppose now . Then is bounded in the lower half-plane by . Let be the reflection of with respect to the real axis, oriented clockwise.
By Residue Theorem, we have
A calculation shows that the residue equals to , so
As , the contribution to the last integral from the semicircle tends to 0 since on the semicircle, giving
For , the integral is elementary as shown below
as . Thus .
Problem 2. Evaluate the integral
Solution. For , the integral is elementary and finally we get . For , the function
is bounded in the upper half-plan and in fact is there. In this case, one has for all . For , by Residue Theorem,
Finally, we obtain
Similar question. Estimate the limit
for a real number .
Hint. For , the limit equals which I had considered here. For , we need to calculate the following residue
which equals by using the following helpful formula
where and are holomorphic and is a simple pole.
The final answer is