I have studied a complex analysis with textbook "Function Theory of One Complex Variable" by Robert Everist Greene, Steven George Krantz.
While studying about "Conformality" and several exercises regarding it, I got a question.
The question is that :
Assume that $F : U \rightarrow \mathbb{C}$ is $C^1$ on an open set $U$.
Let $P$ be in $U$. Assume that $F$ "stretches equally in all directions at $P$". (The size of all directional derivatives is the same for all unit modulus.) Prove that either $F'(P)$ exists or $\bar{F'}(P)$ exists
My attempts to solve it are followings :
Let $u$ and $v$ are a real and imaginery part functions of $F$, repectively. Then I only need to show that these function satify the Cauchy-Riemann Equations. Then it induces that the comlex valued $C^{1}$ function $F$ satisfies Cauchy Riemann equations, so it is holomorphic at $P$ and differentiable at $P$ obviously.
But, with the given informations, I dont know how to induce the Cauchy Riemann Equations. I need some proper advice for my studies.
Best Answer
We use the following easy fact: assume $w,z$ complex numbers s.t. $|z\cos \theta + w\sin \theta |$ doesn't depend on $\theta$, then $w= \pm iz$
(proof: wlog assume $w,z \ne 0$ as otherwise statement trivial and applying this with $\theta=0, \frac{\pi}{2}$ we get $|w|=|z|$ so $w=\alpha z, |\alpha|=1$; but then squaring, $| \cos \theta+ \alpha\sin \theta |^2=1+2(\sin \theta \cos \theta)\Re \alpha$ obviously doesn't depend on $\theta$ iff $\Re \alpha =0$, hence $\alpha =\pm i$ as required)
Applying this with $z=f_x(P), w=f_y(P)$ we get from the hypothesis and the fact above that $f_y(P) =\pm i f_x(P)$. This clearly implies $f$ analytic at $P$ if we have the plus sign and $f$ conjugate analytic if the sign is minus, so we are done !