Definite Integrals of Powers of Trigonometric Functions

三角函数幂的定积分 苏剑林 · 2013-12-05 · Mathematics integral trigonometric function

Lately my main focus has been on path integrals — I've been trying to derive a new transformation method (or perhaps a new perspective) for path integrals, but there's a hurdle I still can't get past, which is why the blog has been updated so sparingly. Speaking of integration and differentiation: these two are originally inverse operations, but under the unifying framework of complex numbers, they can actually be converted into one another. For instance, the Schrödinger equation is the differential form of quantum mechanics, while the path integral can be said to be the integral form of quantum mechanics. This makes me wonder — does every piece of mathematics expressed in differential form have a corresponding integral-form version? And if so, which is superior, the differential version or the integral version?

In mathematical analysis, we tend to feel that differentiation is much easier than integration — there are ready-made formulas for derivatives, and so on. But differentiation has one major drawback: it is multi-component. For example, a potential function is a scalar, but after differentiation (taking the gradient) it becomes a three-component vector (namely, force). Having multiple components makes things genuinely harder to handle, and to deal with this a large number of operators had to be introduced. The virtue of integration lies in its scalar nature — the computation may be complicated, but the underlying idea is easy to grasp. I prefer theories expressed in integral form (such as the principle of least action, the path integral, etc.).

Speaking of common and famous definite integrals in mathematical analysis, I have to mention the following trigonometric integral.

$$\int_0^{\pi/2} \sin^{2n} \theta d\theta$$

It's not hard to show that it is also equal to

$$\int_0^{\pi/2} \cos^{2n} \theta d\theta$$more

There are many ways to evaluate this definite integral. The method given in textbooks proceeds by mathematical induction with a recursive formula, but I find a simpler method — one that's easy to remember (meaning that even if you forget it, you can quickly re-derive it) — is to use the exponential form of trigonometric functions. It suffices to use $\cos\theta=\frac{e^{i\theta}+e^{-i\theta}}{2}$, which lets us rewrite the integral as

$$\begin{aligned} \int_0^{\pi/2} \cos^{2n} \theta d\theta &=\int_0^{\pi/2} \left(\frac{e^{i\theta}+e^{-i\theta}}{2}\right)^{2n} d\theta\\ &=\frac{1}{2^{2n}}\int_0^{\pi/2} \left(e^{i\theta}+e^{-i\theta}\right)^{2n} d\theta \end{aligned}$$

Consider $\left(e^{i\theta}+e^{-i\theta}\right)^{2n}$; it must take the form

$$\frac{(2n)!}{(n!)^2}+\sum_{k=-n,k\neq 0}^{n} a_k e^{2ik\theta}$$

(one could work out the explicit form of the coefficient $a_k$, but that's not important — all we need to know is that it is real). And

$$\int_0^{\pi/2} e^{2ik\theta}d\theta=\left. -i\frac{1}{2k}e^{2ik\theta} \right|_0^{\pi/2}=\frac{[(-1)^{k+1}+1]i}{2k}$$

In fact, we don't even need to compute the value of this integral; we only need to establish that it is purely imaginary. Then

$$\begin{aligned}\int_0^{\pi/2}\cos^{2n} \theta d\theta &=\frac{1}{2^{2n}}\int_0^{\pi/2} \left[ \frac{(2n)!}{(n!)^2} +\sum_{jk=-n,k\neq 0}^{n} a_k e^{2ik\theta}\right]d\theta\\ &=\frac{1}{2^{2n}}\frac{(2n)!}{(n!)^2}\frac{\pi}{2}+i\sum\limits_{k=-n,k\neq 0}^{n}\frac{[(-1)^{k+1}+1]a_k}{2k} \end{aligned}$$

Regardless of what form $a_k$ takes, after summing it and multiplying by $i$, the result must be purely imaginary. But the left-hand side is real, so the imaginary terms must vanish. Hence

$$\begin{aligned}\int_0^{\pi/2}\cos^{2n} \theta d\theta &=\frac{1}{2^{2n}}\frac{(2n)!}{(n!)^2}\frac{\pi}{2}\\ &=\frac{(2n)!}{(2n!!)^2}\frac{\pi}{2}\end{aligned}$$

However, if one tries to use the same trick to compute $\int_0^{\pi/2}\sin^{2n+1}\theta d\theta$ and $\int_0^{\pi/2}\cos^{2n+1}\theta d\theta$, it turns out not to be so simple. In fact, for odd powers, there doesn't seem to be a comparably simple and direct method.