Math 6720 • Applied Complex Variables and Asymptotic Methods

Recall the binomial expansion
\[
(a+b)^n=\sum_{k=0}^n \binom{n}{k}a^{n-k}b^k,
\]
where
\[
\binom{n}{k}=\frac{n!}{k!(n-k)!}.
\]
\begin{enumerate}
\item
(12 points) Use the binomial expansion and Cauchy's Integral Theorem to evaluate
\[
\oint_C \left(z+\frac{1}{z}\right)^{2n}\frac{dz}{z}
\]
where \(C\) is the unit circle centered at the origin.
\item
(13 points) Use this result to establish the following real integral formula:
\[
\frac{1}{2\pi}\int_0^{2\pi}(\cos\theta)^{2n}\,d\theta
=\frac{(2n)!}{4^n(n!)^2}.
\]
\end{enumerate}

\item