10:31

Calculus for Scientists and Engineers: Early Transcendental

Cauchy-Riemann equations has the form $f(x, y)=u(x, y)+i v(x, y),$ where $u$ and $v$ are real-valued functions and $i=\sqrt{-1}$ is the imaginary unit. A function $f=u+i v$ is said to be analytic (analogous to differentiable) if it satisfies the Cauchy-Riemann equations: $u_{x}=v_{y}$ and $u_{y}=-v_{x}$

a. Show that $f(x, y)=\left(x^{2}-y^{2}\right)+i(2 x y)$ is analytic.

b. Show that $f(x, y)=x\left(x^{2}-3 y^{2}\right)+i y\left(3 x^{2}-y^{2}\right)$ is analytic.

c. Show that if $f=u+i v$ is analytic, then $u_{x x}+u_{y y}=0$ and $v_{x x}+v_{y y}=0$