Find the average value of $x^2 - y^2 + 2y$ over the circle $|z - 5 + 2i| = 3$.

Question

Find the average value of $x^2 - y^2 + 2y$ over the circle $|z - 5 + 2i| = 3$.

Could someone please explain how to do this. I keep getting an answer of 17, but my professor says that is incorrect.

Answer 1

First, recognize that the circle has center at $5-2i$ and radius $3$. So you can parametrize it using $x=5+3\cos(t)$ and $y=-2+3\sin(t)$.

So you are now asking for the average value of $$ \begin{align} &(5+3\cos(t))^2-(-2+3\sin(t))^2+2(-2+3\sin(t)) \\ &=17+30\cos(t)+18\sin(t)+9\cos^2(t)-9\sin^2(t) \end{align} $$ where $t$ runs over $[0,2\pi]$. Can you take it from there? Note that $\sin(t)$ and $\cos(t)$ average to $0$, and whatever $\sin^2(t)$ averages out to, it matches what $\cos^2(t)$ averages out to.