Integral of exponential involving sines and cosines

[Repetit]

Hey!

Can someone help me solve the following integral

$$\int_0^{2\pi} exp[-i(G_x \cos\theta + G_y \sin\theta] d\theta$$

I've tried splitting the exponential into a product of two exponetials and rewriting the exponentials in terms of sines and cosines. But I always end up getting stuck. Some of my rewritings ended up looking close to a Bessel function but it's just not quite the same. Can someone just give me a hint on where to start?

Im not sure about the limits, they might have to be from $$-\pi$$ to $$\pi$$ but I don't believe that should change anything.

Thanks in advance

 

[jpr0]

Probably the answer will look like some combination of Bessel Js.

$$J_{n}(x) = \int_{0}^{2\pi}\frac{d\phi}{2\pi}e^{-ix\sin\phi + in\phi}$$

There is an expansion:

$$e^{-ix\sin\phi} = \sum_{k=-\infty}^{\infty}J_{k}(x)e^{-ik\phi}$$

So in principle you can expand each exponentional term into Fourier harmonics and evaluate the angular integral, then resum the resulting expression. There is probably an easier way though...

 

[tacman]

!

Hi there! Solving this integral involves using the Euler's formula, which states that e^(ix) = cos(x) + isin(x). We can rewrite the exponential in the integral as e^(-iG_xcos\theta) * e^(-iG_ysin\theta). Then, using the Euler's formula, we can rewrite these two exponents as cos(G_xcos\theta) + isin(G_xcos\theta) and cos(G_ysin\theta) + isin(G_ysin\theta), respectively. We can then use the product rule for integration to solve this integral. Remember to use the limits of integration as -\pi to \pi, as this will account for all possible values of theta. I hope this helps, good luck with your problem!