오늘의 계산 36 - Integration of a Fejer kernel-like thing

…을 가장한, 옛날에 AoPS에 답변으로 올렸지만 티스토리로 퍼오기 매우 귀찮아서 그냥 방치했던 계산 하나를 올려봅니다. 다른 꼴의 삼각함수 적분에도 쓸 수 있는 테크닉이 아닐까 해서 이렇게 올려봅니다.




Today's integral we are going to evaluate is a famous one,

.

---

Solution 1 (by elementary calculus). It is clear that . To determine for , we consider the difference for . Some trigonometric identities show that

Since

it follows that and for . Therefore for all .

---

Solution 2 (by complex analysis). It is easy to see, by the substitution , that

.

Thus for ,

Since

we have for .

---

Solution 3 (by advanced calculus). Let for . We further assume , for the convenience of calculation. Then

But note that

where is the harmonic number, and logarithmic differentiation of

gives

which implies that

as . Also, for , we find that

Therefore we have



which simplifies to

In special cases, taking for gives .