I'm working through a problem set in Complex Analysis and have encountered the following question:
Problem
Write down the Cauchy-Riemann equations and explain their connection with holomorphic functions. [Completed]
Let $U=\mathbb{C} \diagdown (-\infty , 0]$. Show that for $z=x+iy\in\mathbb{C}$, we may define $u(x,y), \ v(x,y)$ by;
$\sqrt{2}u=\left(x+\sqrt{x^2+y^2} \right)^{\frac{1}{2}}$ $v=\frac{y}{2u}$
where we take positive square roots in the definition of $u$.
I can't see how we can identify a computation for $u$ and $v$ when they're seemingly arbitrary in the question. Is this a misprint, or am I missing something?
Regards,