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I have an integral: $$\int_0^{2\pi}\mathrm{d}\theta e^{ia\cos(\theta-\theta_1)}\cos^2(\theta-\theta_2),$$ where $a, \theta_1$ and $\theta_2$ are reals. Any idea on how to evaluate this integral.

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  • $\begingroup$ This is surely interesting. For the moment, I tried to use Mathematica, and it says that the integral does exist and its result is $$\pi\left(J_0(a) - J_2(a)\cos(2\theta_1 - 2\theta_2)\right)$$ For $a\in\mathbb{R}$. $J_n(z)$ are the Bessel functions of the first kind --> mathworld.wolfram.com/BesselFunctionoftheFirstKind.html I will think something. Maybe you could start with some Taylor series for the cosine? $\endgroup$ – Von Neumann Mar 7 '16 at 21:51
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Let $\theta \mapsto \theta + \theta_1$, so that the integral becomes

$$\frac12 \int_0^{2 \pi} d\theta \, e^{i a \cos{\theta}} \, \left [ 1+\cos{(2 (\theta+\theta_1-\theta_2))} \right ]$$

The first term produces

$$\frac12 \int_0^{2 \pi} d\theta \, e^{i a \cos{\theta}} $$

Let $z=e^{i \theta}$; then the integral is

$$-\frac{i}{2} \oint_{|z|=1} \frac{dz}{z} e^{i (a/2) \left ( z+z^{-1} \right )} $$

To deal with the essential singularity in the exponential, we form the Laurent series of the integrand:

$$-\frac{i}{2} \sum_{n=0}^{\infty} \frac{i^n}{n!} \oint_{|z|=1} \frac{dz}{z} \left ( \frac{a}{2} \right )^{n} \left ( z+\frac1z \right )^n $$

The only nonzero terms will be the constant term of the expansion of the binomial terms, and those only appear in the even terms. Thus, the integral is

$$-\frac{i}{2} i 2 \pi \sum_{n=0}^{\infty} \frac{(-1)^n}{(2 n)!} \binom{2 n}{n} \left ( \frac{a}{2} \right )^{2 n} = \pi \sum_{n=0}^{\infty} \frac{(-1)^n}{(n!)^2} \left ( \frac{a}{2} \right )^{2 n} = \pi \, J_0(a)$$

The next term may be split up as $\cos{(2 (\theta+\theta_1-\theta_2))} = \cos{2 \theta} \cos{2 (\theta_1-\theta_2)} - \sin{2 \theta} \sin{2 (\theta_1-\theta_2)} $.

$$\begin{align} \frac12 \int_0^{2 \pi} d\theta \, e^{i a \cos{\theta}} \cos{2 \theta} &= -\frac{i}{4} \oint_{|z|=1} \frac{dz}{z} e^{i (a/2) \left ( z+z^{-1} \right )} \left (z^2+\frac1{z^2} \right )\\ &= -\frac{i}{4} \sum_{n=0}^{\infty} \frac{i^n}{n!} \oint_{|z|=1} \frac{dz}{z} \left ( \frac{a}{2} \right )^{n} \left ( z+\frac1z \right )^{n} \left (z^2+\frac1{z^2} \right ) \\ &= -\frac{i}{4} i 2 \pi \sum_{n=0}^{\infty} \frac{(-1)^n}{(2 n)!} \left [ \binom{2 n}{n-1} + \binom{2 n}{ n+1}\right ]\left ( \frac{a}{2} \right )^{2 n} \\ &= \pi \sum_{n=0}^{\infty} \frac{(-1)^n}{(n-1)! (n+1)!} \left ( \frac{a}{2} \right )^{2 n} \\ &= -\pi \left ( \frac{a}{2} \right )^{2} \sum_{n=0}^{\infty} \frac{(-1)^n}{n! (n+2)!} \left ( \frac{a}{2} \right )^{2 n} \\ &= -\pi J_2(a)\end{align}$$

Note that the integral involving $\sin{2 \theta}$ has, upon expansion of the exponential, terms such as $\binom{2 n}{n-1} - \binom{2 n}{ n+1}$, which are all zero. Thus,

$$\int_0^{2 \pi} d\theta \, e^{i a \cos{(\theta-\theta_1)}} \, \cos^2{(\theta-\theta_2)} = \pi J_0(a) - \pi J_2(a) \cos{2 (\theta_1-\theta_2)} $$

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  • $\begingroup$ You know when I was just browsing around and saw this integral I thought to myself "I don't have time to do this one but, man this looks like a fun contour integral, I'll bet RG will have a jab at it". I'm impressed by this solution. $\endgroup$ – DaveNine Mar 8 '16 at 17:49
  • $\begingroup$ @DaveNine: Thank you; yes, I find working with essential singularities kind of fun. $\endgroup$ – Ron Gordon Mar 8 '16 at 18:12

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