# Proof Cauchy-Goursat

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

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

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

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

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

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

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

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