- #1

JPaquim

- 34

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## Homework Statement

Show that [itex]\int_{\gamma}\ f^*(z)\ f'(z)\ dz[/itex] is a pure imaginary for any piecewise smooth closed curve [itex]\gamma[/itex] and any [itex]C^1[/itex] function [itex]f[/itex] whose domain contains an open set containing the image of [itex]\gamma[/itex]

**2. The attempt at a solution**

I have tried to approach it from some different angles. I've tried to Taylor expand everything, to see if some interesting things would happen, but I found none. I've also tried to expand it as such:

[itex]\int_{\gamma}\ f^*(z)\ f'(z)\ dz = \int_{\gamma}\ [(u - iv)(\frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x})](dx + idy) = [/itex]

[itex]\int_{\gamma}\ (u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial x})dx+(u\frac{\partial u}{\partial y} + v\frac{\partial v}{\partial y})dy + i\int_{\gamma}\ (u\frac{\partial v}{\partial x}-v\frac{\partial u}{\partial x})dx+(u\frac{\partial v}{\partial y}-v\frac{\partial u}{\partial y})dy[/itex]

What can I do with this? Is this the way to go?

Cheers