I'm trying to show that

$$\int_{0}^{2\pi}\cos(n\theta-2\sin\theta)d\theta = 2\pi\sum_{r=0}^{\infty}\frac{(-1)^n}{r!(n+r)!}$$

The question hints that I should consider


but I don't see how to get to the answer. Considering the nature of the integral, the unit circle seems like a sensible contour. The integrand has only one pole in the unit circle (at zero), which has residue $-1$. So:

$$\int e^{z-z^{-1}}dz=-2\pi i$$

Switching to $\theta$:

$$\int_0^{2\pi}ie^{i\theta}e^{e^{i\theta}-e^{-i\theta}}d\theta=-2\pi i$$


That integrand looks similar, but doesn't match up with the required integral, and I don't see how to get them to match. More troublesome, the required infinite sum on the right hand side is missing.

I've considered switching to a contour of radius $n$, but that just gives me $(2n+1)\pi$ on the right-hand side, and I still can't get it to match on the left. How do I solve this?

EDIT: Infinite sum corrected.

  • 1
    $\begingroup$ It seems you have a slight notational problem, as '$n$' is a dummy index on the RHS, while it is a specific number (integer?) on the LHS... $\endgroup$ Dec 23 '16 at 17:24
  • $\begingroup$ This integral is related to an integral representation of Bessel function $J_n$. See formula (10.9.2) in (dlmf.nist.gov/10.9) with $z=2$ $\endgroup$
    – Jean Marie
    Dec 23 '16 at 17:56
  • $\begingroup$ Infinite sum corrected. @JeanMarie's hint about the Bessel function brought me to: math.stackexchange.com/questions/1051558/…, which solves a more general case of this. $\endgroup$
    – Number 34
    Dec 23 '16 at 23:00
  • $\begingroup$ Rewriting the whole thing in terms of complex exponentials gives me $\int_0^{2\pi}\cos(n\theta-2\sin\theta)d\theta &= \Re\int_0^{2\pi}e^{in\theta-2i\sin\theta}d\theta = \Re\int_0^{2\pi}\left(e^{i\theta}\right)^n\exp\left(-e^{i\theta}+e^{-i\theta}\right)d\theta=\Re\oint_{|z|=1}z^ne^{-\left(z-\frac{1}{z}\right)}dz$, which contains the quantity you were given as a hint and also a dependancy on $n$. Can you take it from there? $\endgroup$
    – Tom
    Dec 24 '16 at 0:03
  • $\begingroup$ Sorry for that. Now I can't even click on the edit button, can anybody help with that? $\endgroup$
    – Tom
    Dec 24 '16 at 0:08

The integral is the real part of

$$\int_0^{2 \pi} d\theta \, e^{i n \theta} \, e^{-i 2 \sin{\theta}} $$

To convert this into a complex integral over the unit circle, sub $z=e^{i \theta}$ and the integral is

$$-i \oint_{|z|=1} dz \, z^{n-1} \, e^{-\left ( z-z^{-1} \right )} $$

This integrand has an essential singlarity at $z=0$. To evaluate the integral using the residue theorem, we replace the exponential by its series representation and reverse order of sum and integral. The integral is then equal to

$$-i \sum_{k=0}^{\infty} \frac{(-1)^k}{k!} \oint_{|z|=1} dz \, z^{n-1} \, \left ( z-z^{-1} \right )^k$$

We can then expand the binomial inside the integral:

$$-i \sum_{k=0}^{\infty} \frac{(-1)^k}{k!} \oint_{|z|=1} dz \, z^{n-1} \, \sum_{j=0}^k (-1)^{k-j} \binom{k}{j} z^{2 j-k}$$

Note that the integral is going to be zero unless $k \ge n$. To help us see this a bit better, let's shift the sum to begin there:

$$-i \sum_{k=0}^{\infty} \frac{(-1)^{n+k}}{(n+k)!} \oint_{|z|=1} dz \, z^{-1} \sum_{j=0}^{n+k} (-1)^{k-j} \binom{n+k}{j} z^{2 j-k}$$

The residue is then the coefficient of $z^0$ inside the sum. The exponent of $z$ is zero when $j=k/2$, so only even values of $k$. Let's clean this up a bit by only summing over even value of $k$:

$$-i \sum_{k=0}^{\infty} \frac{(-1)^{n+2 k}}{(n+2 k)!} \oint_{|z|=1} dz \, z^{-1} \sum_{j=0}^{n+2 k} (-1)^{2 k-j} \binom{n+2 k}{j} z^{2 j-2 k}$$

Now we see that the exponent of $z$ is zero when $j=k$. Thus, we may now apply the residue theorem and the integral is equal to

$$2 \pi \sum_{k=0}^{\infty} \frac{(-1)^{n}}{(n+2 k)!} (-1)^k \binom{n+2 k}{k}$$

Simplifying, and noting the the integral is entirely real, we get that

$$\int_0^{2 \pi} d\theta \, \cos{\left ( n \theta - 2 \sin{\theta} \right )} = 2 \pi \sum_{k=0}^{\infty} \frac{(-1)^{n+k}}{k! (n+k)!}= 2 \pi (-1)^n J_n(2)$$

I believe my answer differs from that of the OP by a factor of $(-1)^k$ in the sum.

  • $\begingroup$ really a pity that such a nice answer recieves absolutly no attention (not even from the OP)..(+1) from me anyways $\endgroup$
    – tired
    Jan 7 '17 at 13:50

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