# $\frac{\omega}{2\pi}\int_0^{\frac{2\pi}{\omega}}\frac{\sin^2\theta \cos^2\omega t}{(1+\beta\cos\theta \sin\omega t)^5} dt$

I'm going to write out the whole problem as it is given to me (bad grammar and all) even though some of the info may be irrelevant to finding a solution.

A charge $e$ moving along a straight line undergoes simple harmonic motion with frequency $\omega$ and amplitude $a$. Since the charge is moving with an acceleration, it emits electromagnetic radiation. The angular distribution of the instantaneous (time-dependent) intensity of the radiation is $$I(\theta,t)=\kappa \frac{\sin^2\theta \cos^2\omega t}{(1+\beta\cos\theta\sin\omega t)^5}$$ where $\theta$ is the angle between the direction of the oscillations and the direction of the emission, $\kappa$ is an irrelevant for us constant, $$\beta=\frac{\omega a}{c},0\leq\beta<1$$ is the ratio of the maximal speed of the charge to the speed of light. Find the angular distribution of the averaged intensity of the emission, $$I(\theta)=\frac{\omega}{2\pi}\int_0^{\frac{2\pi}{\omega}}I(\theta,t)dt$$ Use contour integration and use a computer algebra system to calculate the relevant residue(s).

I'm not even sure how to start this problem. If it only had $\cos$ or $\sin$ terms i would Use Euler's to rewrite in terms of the Real or Imaginary parts of $e^{i\theta}$ but with with these two different trig functions i'm not sure how i would do that. Another problem I'm having is understanding what the problem is even asking. It wants me to find $I(\theta)$ but it wants me to integrate over $t$. Does that mean my solution will be in terms of $\theta$? In which case i can factor out the $\sin^2\theta$ from the numerator? About the only thing i'm remotely sure about is that after i somehow convert this to a function $f(z)$ I'll have to find the singularity(s) enclosed within the integration contour and use Residues to determine a solution. But i'm not even sure how to begin converting this to a usable function $f(z)$.

• @YuriyS - I understand how changing the variable will make the limits a bit nicer but i'm not sure i understand how to change the variable to $u=\tan(\omega t/2)$ and how that would help me. Also, what would i do with that $\cos\theta$ in the demoninator? I can't factor it out like i did with the sine term in the numerator. – Iron Charioteer Apr 24 '16 at 22:14
• Yes, i've finished calculus courses but i haven't done any complex variable math except in this mathematical methods class. So you're saying treat this like a double integral where i integrate with respect to t and treat all the $\theta$ terms like a constant? – Iron Charioteer Apr 24 '16 at 22:26
• Basically yes, but there is not need to think of this as a double integral. Just treat $\theta$ like a constant, as you said. If it is required of you to use contour integration see here the general procedure for rational combinations of trig functions – Yuriy S Apr 24 '16 at 22:32
• That last link looks more like the route i'm supposed to take. Switching trig functions into exponentials and such. Let me see what i can do working from that link. Thank you. – Iron Charioteer Apr 24 '16 at 22:40
• So i've managed to convert the integral to $I(\theta)=\frac{\sin^2\theta}{8\pi i}\int_C\frac{z^4+2z^2+1}{[1+\beta\frac{\cos\theta}{2i}(z-\frac{1}{z})]^5}\frac{dz}{z^3}$ and i'm not sure how to handle the denominator. I know i have to find the singularity but i'm not sure how to go about doing that. Seems that doing a 5th order binomial expansion would only make things more complicated. I must be missing some obvious algebra trick to make this easier.... – Iron Charioteer Apr 24 '16 at 23:26