Let $\gamma\colon[a,b]\to \mathbb{C}$ denote a piecewise differentiable path , and let $\varphi:$ Image $\gamma\colon \to \mathbb{C}$ be a continuous function.
Define $g: D = \mathbb{C}$-Image$\varphi \to \mathbb{C}$ by: $ g\left( z \right) = \int\limits_\gamma {\frac{{\varphi \left( u \right)}} {{u - z}}du}. $
Prove that: $ g^{\left( n \right)} \left( z \right) = n!\int\limits_\gamma {\frac{{\varphi \left( u \right)}} {{\left( {u - z} \right)^{n + 1} }}du}. $
Well , it's obvious that is enough to prove only the case $n=1$. Computing $ \eqalign{ & \frac{{g\left( {z + h} \right) - g\left( z \right)}} {h} = \int\limits_\gamma {\frac{{\varphi \left( u \right)}} {{\left( {u - z - h} \right)}} - \frac{{\,\varphi \left( u \right)}} {{u - z}}du = } \cr & \int\limits_\gamma {\varphi \left( u \right)\frac{h} {{\left( {u - z - h} \right)\left( {u - z} \right)}}du} \cr} $ Well I have no idea how to continue.