- #1

GwtBc

- 74

- 6

## Homework Statement

Prove that the function ## f(z)= 1/\sqrt{2}(\sqrt{\sqrt{x^{2}+y^{2}}+x}+i*sgn(y)\sqrt{\sqrt{x^{2}+y^{2}}-x})## is holomorphic on the domain ## \Omega = \left \{ z: z \neq 0, \left | \arg{z} \right | <\pi\right \} ## and further that in this domain ##f(z)^{2} = z. ##

## Homework Equations

if

## f(z) = u(x,y) + iv(x,y) ##

Then we have the Cauchy-Riemann relations for a holomorphic function

## \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y},\hspace{0.3cm} \frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y}##

## The Attempt at a Solution

Essentially I've been trying to show that the function conforms with the Cauchy Riemann equations, but there are two issues. Firstly, the imaginary component seems not to be differentiable where y = 0. I tried invoking first principles but the limit doesn't exist. Second is that it's not immediately obvious that the Cauchy relations hold even where im{f} is differentiable. I could see this latter issue being resolved by some algebraic trick, but I'm kind of clueless about the first.

Thanks in advance everyone.