I cannot understand the following statement in the proof of Lemma 8.7 in "Lectures on Riemann Surfaces" by Otto Forster.

whereQuote:

$\displaystyle \phi(z)=\frac{1}{2 \pi i}\int_{|w-w_0|=\epsilon}w\frac{F_w(z,w)}{F(z,w)}dw, \;\;\; \left( F_w := \frac{\partial F}{\partial w}\right)$

Since the integral depends holomorphically on $\displaystyle z$, the function $\displaystyle z\mapsto \phi(z)$is holomorphic on $\displaystyle D(r)$.

$\displaystyle D(R)=\{z\in C : |z|<R \}, \;\;\; R>0,$

$\displaystyle F(z,w)=w^n+c_1(z)w^{n-1}+\cdots+c_n(z),$

$\displaystyle c_1(z)\cdots c_n(z)$are holomorphic functions on $\displaystyle D(r)$.

In the equation

$\displaystyle \phi(z)-\phi(z_0)=\frac{1}{2 \pi i}\int_{|w-w_0|=\epsilon}w\left(\frac{F_w(z,w)}{F(z,w)}-\frac{F_w(z_0,w)}{F(z_0,w)}\right)dw$

the integrand can be written as$\displaystyle (z-z_0)P(z,w)$ where $\displaystyle P(z,w)$ is holomorphic with respect to $\displaystyle z$, to prove $\displaystyle \phi(z)$ to be holomorphic, it seems enough to show that $\displaystyle \int_{|w-w_0|=\epsilon}P(z,w) dw$ is continuous with respenct to $\displaystyle z$. But, I don't know how to show it.

Any help would be appreciated.

Thanks in advance.