Proof Cauchy-Goursat

**manjohn12** · March 12th 2009, 08:18 PM

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?

**ThePerfectHacker** · March 12th 2009, 08:47 PM

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.

**manjohn12** · March 12th 2009, 08:49 PM

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 $f_x = g_x +ih_x$ ?

**ThePerfectHacker** · March 12th 2009, 08:50 PM

Originally Posted by manjohn12

You mean $f_x = g_x +ih_x$ ?

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

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