2018-12.md

course	course_year	question_number	tags	title	year
Complex Methods	IB	12	IB 2018 Complex Methods	Paper 3, Section I, A	2018

(a) Let $f(z)=\left(z^{2}-1\right)^{1 / 2}$. Define the branch cut of $f(z)$ as $[-1,1]$ such that

$$f(x)=+\sqrt{x^{2}-1} \quad x>1$$

Show that $f(z)$ is an odd function.

(b) Let $g(z)=\left[(z-2)\left(z^{2}-1\right)\right]^{1 / 2}$.

(i) Show that $z=\infty$ is a branch point of $g(z)$.

(ii) Define the branch cuts of $g(z)$ as $[-1,1] \cup[2, \infty)$ such that

$$g(x)=e^{\pi i / 2} \sqrt{|x-2|\left|x^{2}-1\right|} \quad x \in(1,2) .$$

Find $g\left(0_{\pm}\right)$, where $0_{+}$denotes $z=0$ just above the branch cut, and $0_{-}$denotes $z=0$ just below the branch cut.