오늘의 계산 40 - A Bizarre Integral

Today we are going to prove the following equality.

Problem. Show that

Proof. Let denote this integral hereafter.

To achieve this, we first observe by double-angle formula that

Here, we used the transformation for each step. Then by symmetry, this is written as

Then by the identity

we have

Then we make change of variable to obtain

where denote the counterclockwised unit circle centered at the origin. Now let denote the innermost integrand of the above equality. To apply Cauchy integration formula, we have to find the pole inside the circle . So let and denote

respectively. Note that and are zeros of quadratic polynomials

and ,

respectively, so that

Now for , only and are contained in the interior of the disk bounded by . Thus by Cauchy integration formula, we have

To evaluate the integral in the right hand side, we note that

for , hence by analytic continuation, also for and . Then

뭐랄까, 이번에는 제가 자주 쓰던 테크닉 이외에도 복소 테크닉을 섞어서 계산해봤습니다. 하지만 무엇보다도 이번 계산의 백미는 맨 마지막에 아크탄젠트 값들이 점차 붕괴(?)되어서 결국 깔끔한 결과로 귀결되는 부분이지요.

그나저나 요즘 통 포스팅을 못해서 그런지 손님들이 줄고 있다는 사실을 체감합니다. 좀 더 분발해야겠어요.