# Proof Cauchy-Goursat

• Mar 12th 2009, 08:18 PM
manjohn12
Proof Cauchy-Goursat
Cauchy-Goursat Theorem: If $f(z)$ is analytic inside and on a simple, closed piecewise smooth curve $C$, then $\oint_{C} f(z) \ dz = 0$.

In proving the "special case" where we use Greens Theorem and the Cauchy Riemann Equations, is assuming that $f$ is continuously differentiable the same thing as saying that $f$ has continuous partial derivatives?
• Mar 12th 2009, 08:47 PM
ThePerfectHacker
Quote:

Originally Posted by manjohn12
Cauchy-Goursat Theorem: If $f(z)$ is analytic inside and on a simple, closed piecewise smooth curve $C$, then $\oint_{C} f(z) \ dz = 0$.

In proving the "special case" where we use Greens Theorem and the Cauchy Riemann Equations, is assuming that $f$ is continuously differentiable the same thing as saying that $f$ has continuous partial derivatives?

If $f$ is continous differenciable (on some open set) it means $f$ is differenciable and $f'$ is continous (on this open set). Write $f = g + hi$. Remember that $f ' = g_x + h_x i$. Thus, $g_x,h_x$ are continous on this open set. However, by Cauchy-Riemann equations $g_x=h_y$ and $h_x = -g_y$ so $h_y,g_y$ are continous also. Thus, the partial derivatives must be continous.
• Mar 12th 2009, 08:49 PM
manjohn12
Quote:

Originally Posted by ThePerfectHacker
If $f$ is continous differenciable (on some open set) it means $f$ is differenciable and $f'$ is continous (on this open set). Write $f = g + hi$. Remember that $f ' = g_x + h_x i$. Thus, $g_x,h_x$ are continous on this open set. However, by Cauchy-Riemann equations $g_x=h_y$ and $h_x = -g_y$ so $h_y,g_y$ are continous also. Thus, the partial derivatives must be continous.

You mean $f_x = g_x +ih_x$?
• Mar 12th 2009, 08:50 PM
ThePerfectHacker
Quote:

Originally Posted by manjohn12
You mean $f_x = g_x +ih_x$?

No. The derivative of $f$ is computed to be $g_x + ih_x$.