<s>[INST] Angular integrals used in QED I am reading a research paper
 and am stuck at a point where the author uses angular integrals.  I don't have any idea about it and would like help. 
 The angular integral is:
$$I_k (y)=\int_0^\pi  {\sin^2(\theta)\cos^{2k}(\theta) \over \beta^2(y)-\cos^2 (\theta)} d\theta$$
The author directly give results to this integral without proof.
The results are:
$$I_0 (y)=\pi \left(1-\sqrt{1-{1\over\beta^2 (y)} }  \right),$$
$$I_1 (y)={-\pi \over 2}+\beta^2 (y)I_0 (y),$$
I don't know how he got these results.  Can anyone please help me?
 [/INST] An interesting approach to these integrals is the use of complex integration. In this case, we can use the fact that
with
\begin{equation}
f(\sin\theta,\cos\theta) = \frac{\sin^2\theta \cos^{2k} \theta}{\beta^2-\cos^2\theta}
\end{equation}
we have
\begin{eqnarray}
\int_0^\pi \mathrm{d}\theta \, f(\sin\theta,\cos\theta) &=&
\frac{1}{2} \int_0^{2\pi} \mathrm{d}\theta \,f(\sin\theta,\cos\theta)
\\
&=& \frac{1}{2} \int_{|z|=1} \frac{\mathrm{d} z}{i z} \, f\left(\frac{z-z^{-1}}{2i},\frac{z+z^{-1}}{2}\right)
\\
&=&  - \frac{i}{2^{2k+1}} \,\int_{|z|=1} \mathrm{d} z\, \frac{(z^2-1)^2 (z^2+1)^{2k}}{z^{2k+1} (z^4-4 \beta^2 z^2+2z^2+1)} 
\\
&=&  - \frac{i}{2^{2k+1}} \,\int_{|z|=1} \mathrm{d} z\, \frac{(z^2-1)^2 (z^2+1)^{2k}}{z^{2k+1} \Pi_{i=1}^4 (z-z_i)}  \, ,
\end{eqnarray}
with $z_i$ the simple poles
\begin{eqnarray}
z_1 &=& -\beta - \sqrt{\beta^2 - 1}
\\ 
z_2 &=& \beta - \sqrt{\beta^2 - 1}
\\ 
z_3 &=& -\beta + \sqrt{\beta^2 - 1}
\\ 
z_4 &=& \beta + \sqrt{\beta^2 - 1} \,.
\end{eqnarray}
By looking at the result, seems like the author assumes the range of $\beta$ is such that only $z_2$ and $z_3$ lay inside the unit circle (besides the $(2k+1)$-order pole at the origin). Add the contributions from those 3 poles and you can actually find a formula for arbitrary $k$ with relative ease.
More precisely, if we write
\begin{equation}
g = \frac{(z^2-1)^2 (z^2+1)^{2k}}{z^{2k+1} \Pi_{i=1}^4 (z-z_i)} \,
\end{equation}
then we find
\begin{eqnarray}
\int_0^\pi \mathrm{d}\theta \, f(\sin\theta,\cos\theta) = && \frac{\pi}{2^{2k}} \left\{ \frac{1}{(2k)!} \lim_{z\rightarrow 0} \frac{d^{2k}}{dz^{2k}} \left[z^{2k+1} g\right] + \lim_{z\rightarrow z_2} \left[(z-z_2)g\right] + \lim_{z\rightarrow z_3} \left[(z-z_3)g\right] \right\} \, .
\end{eqnarray}
Replacing and simplifying gives us
\begin{equation}
I_k(y) = \frac{\pi}{2^{2k}} \frac{1}{(2k)!} \lim_{z\rightarrow 0} \frac{d^{2k}}{dz^{2k}} \left[ \frac{(z^2-1)^2 (z^2+1)^{2k}}{z^4+z^2\left[2-4\beta^2(y)\right] + 1} \right] - \pi \beta^{2k-1}(y) \sqrt{\beta^2(y) - 1} \, .
\end{equation}
Finally, when expanding the first term, it turns out this can be further simplified, producing
\begin{equation}
I_k(y) = - \pi \sum_{m=0}^k \frac{(2m-3)!!}{(2m)!!} \beta^{2k-2m}(y) - \pi \beta^{2k-1}(y) \sqrt{\beta^2(y) - 1} \, ,
\end{equation}
where we take $(-3)!! = -1$, $(-1)!! = 1$ and $0!! = 1$. This also allows us to generalize the relation between successive values of $I_k(y)$, where we have
\begin{equation}
I_k(y) = - \frac{(2k-3)!!}{(2k)!!} \pi + \beta^{2}(y) I_{k-1}(y) \,.
\end{equation}
In hindsight, this result could have been deduced in a couple of lines, by noticing that the difference between the integrands of $I_k(y)$ and $\beta^2(y) I_{k-1}(y)$ simplifies to
\begin{equation}
\frac{\sin^2\theta \cos^{2k} \theta}{\beta^2(y)-\cos^2\theta} - \beta^2(y) \frac{\sin^2\theta \cos^{2k-2} \theta}{\beta^2(y)-\cos^2\theta} = \cos^{2k} \theta - \cos^{2k-2} \theta \,.
\end{equation}
Hence
\begin{eqnarray}
I_k(y) - \beta^2(y) I_{k-1}(y) &=& \int_0^\pi \mathrm{d}\theta \left[ \cos^{2k} \theta - \cos^{2k-2} \theta \right]
\\
&=& \pi \left[ \frac{(2k-1)!!}{(2k)!!} - \frac{(2k-3)!!}{(2k-2)!!} \right]
\\
&=& - \frac{(2k-3)!!}{(2k)!!} \pi \, ,
\end{eqnarray}
for $k \geq 1$.
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