# Contour Integration

• Apr 10th 2009, 07:21 AM
Contour Integration
Here's a question a just did without any problems and the given solution for it.

Evaluate $\displaystyle \int_C |z|^2 dz$ if C is the square with vertices at (0, 0), (1, 0), (1, 1), (0, 1) (traversed anti-clockwise, so that the vertices come in this order).

(Solution is the attachment)

Now my question is what would happen if it wasn't a square but a rectangle with vertices at (1,1), (2,1), (2,3), (1,3). How would i parametrize z?

Here's what i thought it might be but I'm pretty sure its wrong, or at least some of them are. (You'll have to look at the given solution to see what AB, etc, represents)
AB -> (i + t + 1) {perhaps should just be (i + t)}
BC -> (2 + 2it)
CD -> (3i + 2t)
DA -> (1 + 2it)

• Apr 10th 2009, 09:34 AM
NonCommAlg
Quote:

Here's a question a just did without any problems and the given solution for it.

Evaluate $\displaystyle \int_C |z|^2 dz$ if C is the square with vertices at (0, 0), (1, 0), (1, 1), (0, 1) (traversed anti-clockwise, so that the vertices come in this order).

(Solution is the attachment)

Now my question is what would happen if it wasn't a square but a rectangle with vertices at (1,1), (2,1), (2,3), (1,3). How would i parametrize z?

Here's what i thought it might be but I'm pretty sure its wrong, or at least some of them are. (You'll have to look at the given solution to see what AB, etc, represents)
AB -> (i + t + 1) {perhaps should just be (i + t)}
BC -> (2 + 2it)
CD -> (3i + 2t)
DA -> (1 + 2it)

in general any line segment, say from $\displaystyle P(a,b)$ to $\displaystyle Q(c,d),$ is parametrized by $\displaystyle x=ta+(1-t)c, \ y = tb + (1-t)d, \ 0 \leq t \leq 1,$ or briefly $\displaystyle tP + (1-t)Q, \ 0 \leq t \leq 1,$