1.II.16B

(a) Show that if $f$ is an analytic function at $z_{0}$ and $f^{\prime}\left(z_{0}\right) \neq 0$, then $f$ is conformal at $z_{0}$, i.e. it preserves angles between paths passing through $z_{0}$.

(b) Let $D$ be the disc given by $|z+i|<\sqrt{2}$, and let $H$ be the half-plane given by $y>0$, where $z=x+i y$. Construct a map of the domain $D \cap H$ onto $H$, and hence find a conformal mapping of $D \cap H$ onto the disc $\{z:|z|<1\}$. [Hint: You may find it helpful to consider a mapping of the form $(a z+b) /(c z+d)$, where ad $-b c \neq 0$.]

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