Having Fun with an Integral

The main point is not quantity or speed—the main point is quality of thought.

–- Jane Gilman

$$I = \int_{0}^{2\pi} e^{\cos \theta} \cos(\sin \theta)\; d\theta$$

is an integral that WolframAlpha cannot YET compute analytically as of today (December 8, 2018). Here is the link.

I think that there are two ways to tackle this problem by exploiting the analyticity of the integrand $e^{\cos \theta} \cos(\sin \theta)$. We can calculate it in the real domain by Taylor series or in the complex domain by Cauchy's integral formula.

Solution by Taylor Series

First, let us introduce the function

$$f(r) = \int_{0}^{2\pi} e^{r \cos \theta} \cos(r \sin \theta)\; d\theta,$$

so we have the equation

$$I = f(1)$$

. By Taylor expansion, we have

$$f(r) = \sum_{n=0}^{\infty} \frac {f^{(n)}(0)}{n!} r^{n}.$$

It is easy to show that all the coefficients of this Taylor expansion vanish except for the first one. Namely, $f(0) = \int_{0}^{2\pi} e^{0 \cos \theta} \cos(0 \sin \theta)\; d\theta = 2\pi$. The $n$-th derivatives of $f$ at $0$ must be in the form of

$$f^{(n)}(0) = \int_0^{2\pi} \sum_{k{\text{ even }}\wedge\; 0\le k\le n}(-1)^{\frac {k}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta \; d\theta = \int_{0}^{2\pi} \cos(n \theta) \; d\theta = 0.$$

Hence, we have $f(r) = 2\pi = I$.

Solution by Cauchy's Integral Formula

First, let us introduce the same function

$$f(r) = \int_{0}^{2\pi} e^{r \cos \theta} \cos(r \sin \theta)\; d\theta.$$

The integrand can be converted to a complex exponential,

$$e^{r \cos \theta} \cos(r \sin \theta) = \Re\{e^{r \cos \theta} \cos(r \sin \theta) + i \sin(r \sin \theta)\} = \Re\{e^{r\cos \theta + ir\sin \theta}\} = \Re\{e^{re^{i\theta}}\}.$$

Cauchy's Integral formula at $0$ is

$$g(0)=\frac{1}{2\pi i} \oint_\gamma \frac{g(z)}{z} \,dz.$$

If we take $g(z) = e^z$, and $\gamma: re^{i\theta}$, we have

$$g(0)=\frac{1}{2\pi i} \int_{0}^{2\pi} \frac{g(re^{i\theta}) ire^{i\theta}}{re^{i\theta}} \,d\theta = \frac{1}{2\pi} \int_{0}^{2\pi} g(re^{i\theta}) \,d\theta.$$

Note that we have

$$\Re\{g(re^{i\theta})\} = \Re\{e^{re^{i\theta}}\} = e^{r \cos \theta} \cos(r \sin \theta).$$

Hence,

$$\begin{aligned} g(0) &= e^0 = 1 \\ g(0) &= \frac{1}{2\pi} f(r) \\ f(r) &= 2\pi \\ I &= f(1) = 2\pi. \end{aligned}$$