<s>[INST] Complex Analysis Gauss Mean Value Theorem Have I done this correctly?
Evaluate
$\displaystyle\int_{0}^{2\pi}\sin^3(3e^{i\theta} +\frac{\pi}{4})d\theta$
Gauss MVT:
$$f(z_0)=\frac{1}{2\pi}\displaystyle\int_0^{2\pi}f(z_0+re^{i\theta})d\theta$$

So we have the following:
$$\frac{1}{2\pi}f(z_0)=\frac{1}{2\pi}\displaystyle\int_0^{2\pi} \sin^3(3e^{i\theta} + \frac{\pi}{4})d\theta $$
With,
$$z_0=\frac{\pi}{4}$$ and 
\begin{align*}
f(z_0) &= sin^3(z_0) \\
f(\frac{\pi}{4})&=\frac{1}{2\sqrt2} \\
\frac{1}{2\pi}f(z_0)&=\frac{1}{2\pi}\cdot \frac{1}{2\sqrt2} \\
&= \frac{1}{4\sqrt{2}\pi}\\
\end{align*}
Thanks.
 [/INST] Since $\sin^3 z$ is a holomorphic function, its average over a circle centered at $\pi/4$ is equal to $\sin^3(\pi/4)=2^{-3/2}$. The average is $\frac{1}{2\pi}\int_0^{2\pi}\dots$, therefore 
$$ \int_{0}^{2\pi}\sin^3(3e^{i\theta} +\frac{\pi}{4})\,d\theta = 2\pi \cdot 2^{-3/2}$$
</s>