Recall that a function $f(z)$ is called radial if it is constant along the circles of center $0$. Let $f$ be a radial holomorphic function defined on the unit disc $D$.
Show that $f$ is constant. (Hint: apply the
Cauchy-Riemann equations)
I've come across this problem but I've never heard of radial functions.
I know that if $f$ is holomorphic then it is complex differentiable and so $\frac{\delta f}{\delta x} $ and $\frac{\delta f}{\delta y}$ both exist. And so we have the Cauchy-Riemann equations that hold.
But how do I use these facts (that $f$ is radial and holomorphic) to prove that it's constant ?
Please I need help. Thank you
Best Answer
Use Cauchy Riemann equations in polar coordinates $(r,\theta)$ [ Ref: Cauchy-Riemann equations in polar form. ]
Since the real and imaginary parts depend only on $r$ and not on $\theta$ the partial derivatives of $u =\Re f$ and $v =\Im f$ w.r.t. $\theta$ are $0$. But Cauchy Riemann equations now show that the partial derivatives w.r.t. $r$ are also $0$ forcing these functions to be constants.