# Holomorphic function

• Mar 20th 2011, 03:42 AM
EinStone
Holomorphic function
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?
• Mar 20th 2011, 03:51 AM
girdav
Did you compute $\frac{g(z)-g(z_0)}{z-z_0}$ for $z,z_0\in\mathbb C\setminus \overline D$ ?
• Mar 20th 2011, 04:13 AM
EinStone
Why do I do this, to check whether its holomorphic?
• Mar 20th 2011, 04:30 AM
girdav
Yes, we try to check the definition of holomorphic.
• Mar 20th 2011, 08:28 AM
EinStone
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.