Show that $f$ is holomorphic in the unit disk
by Hidenori
Proposition
Define $f: D[0, 1] \rightarrow \mathbb{C}$ through
\[\begin{align*} f(z) = \int_{[0, 1]} \frac{dw}{1 - wz}. \end{align*}\]Prove that $f$ is holomorphic in the unit disk $D[0, 1]$.
Solution
Let $\gamma$ be a closed, piecewise smooth curve in $D[0, 1]$.
\[\begin{align*} \int_{\gamma} f(z) &= \int_{\gamma} (\int_{[0, 1]} \frac{dw}{1 - wz}) dz \\ &= \int_{[0, 1]} (\int_{\gamma} \frac{dw}{1 - wz}) dz \\ &= \int_{[0, 1]} 0 dz & \text{(Corollary 4.13)} \\ &= 0. \end{align*}\]By Corollary 5.6, $f$ is holomorphic.
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