**Problem 1**. Compute

via complex variable methods.

**Problem 2**. Compute

via complex variable methods.

**Problem 3**. Compute

via complex variable methods.

*Solution to problem 2*. To see that the integral exists, notice that

and

as . As the integrand is an even function, we have

.

Let

.

Then is analytic except for a simple pole at 0. For , consider the contour

by Cauchy’s Theorem. We have, by the Residue Theorem,

.

It is easy to see that on we have

since

.

Therefore,

.

Since 0 is the simple pole, then

.

Thus

.

In other word,

.

Finally, since

we have

.

*Solution to problem 3*. The integral exists since it is absolutely integrable with the same argument as above. Since

then

.

The contour is the same to the proof of the problem 2. By Cauchy’s Theorem, one has

.

The integral over tends to 0 as . To estimate the integral over , we note that

which yields

.

Hence we cannot apply the above argument as in the proof of Problem 2 to estimate the integral over . However, this integral can be computed directly as follows

Therefore

as . And thus,

.

*Solution to problem 1*. The integral clearly exists. The proof is similar to the proof of Problem 2. The answer is

.

The solution for Problem 2 is wrong. This integral from 0 to infinity is supposed to be pi/2. As an even function, your answer should be pi.

Comment by Priscila — February 26, 2011 @ 23:58

Thanks for commenting. The proof is correct but there is a coefficient-problem. It shoule be

Comment by Ngô Quốc Anh — February 27, 2011 @ 0:07

Hay!

Comment by TriPhuong — January 25, 2013 @ 18:53

please send me solution of sinmx/x of limit 0 to infinite this question should be solve by residue theorem

Comment by divya — March 21, 2018 @ 19:35