# Contour Integration

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• Apr 10th 2009, 08:21 AM
Deadstar
Contour Integration
Here's a question a just did without any problems and the given solution for it.

Evaluate $\int_C |z|^2 dz$ 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)

Any help on this please?
• Apr 10th 2009, 10:34 AM
NonCommAlg
Quote:

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

Evaluate $\int_C |z|^2 dz$ 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)

Any help on this please?

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