2012-7.md

course	course_year	question_number	tags	title	year
Complex Analysis	IB	7	IB 2012 Complex Analysis	Paper 4, Section I, $4 \mathrm{E}$	2012

Let $h: \mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic function with $h(i) \neq h(-i)$. Does there exist a holomorphic function $f$ defined in $|z|<1$ for which $f^{\prime}(z)=\frac{h(z)}{1+z^{2}}$ ? Does there exist a holomorphic function $f$ defined in $|z|>1$ for which $f^{\prime}(z)=\frac{h(z)}{1+z^{2}}$ ? Justify your answers.