Holomorphic function

**EinStone** · March 20th 2011, 03:42 AM

Hey who can help me with this:
Let f be holomorphic in a neighborhood of a closed disk $\bar{D}$ . Then $g(z) = \int_{\delta D}\frac{f(\zeta)d\zeta}{\zeta -z}$ is a holomorphic function on $\mathbb{C}-\bar{D}$ .
Why, and which one?

**girdav** · March 20th 2011, 03:51 AM

Did you compute $\frac{g(z)-g(z_0)}{z-z_0}$ for $z,z_0\in\mathbb C\setminus \overline D$ ?

**EinStone** · March 20th 2011, 04:13 AM

Why do I do this, to check whether its holomorphic?

**girdav** · March 20th 2011, 04:30 AM

Yes, we try to check the definition of holomorphic.

**EinStone** · March 20th 2011, 08:28 AM

Im somehow lost, when I compute the quotient I just get f'(z) multiplied by a constant. And I dont really use the fact that z is outside D.

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