TITLE: Complex Integration parametrisation
QUESTION [0 upvotes]: I'm trying to integrate $\int_\gamma (z^2-2)dz$ where $\gamma$ is a spiral that loops 3 times and ends at (3,0) on the Argand diagram.
I have found the parametric equations for this contour to be $\left(\frac{3}{2\pi}t\cos(3t),\frac{3}{2\pi}t\sin(3t)\right)$ for $t \in [0,2\pi]$. How do I turn this into a suitable parametrization to integrate?
I considered setting $\gamma (t)=\left(\frac{3}{2\pi}t\cos(3t)\right) +i\left(\frac{3}{2\pi}t\sin(3t)\right)$, but I wasn't sure if this worked?
EDIT: Trying this with the exponential identity for my version of $\gamma=\frac{3e^{3it}t}{2\pi}$ I ended up with $\frac{-39}{8}$ which doesn't seem right...

REPLY [0 votes]: Your path is homotopic to another, very simple path: the line segment between the endpoints.
{"set_name": "stack_exchange", "score": 0, "question_id": 1174757}