Injective Holomorphic Functions that are not Conformal?

Question

Is it possible for a holomorphic function on a connected domain to be conformal but not injective? Also, is it possible for a holomorphic function to be injective but not conformal?

Answer 1

Example. The exponential function is conformal but not injective: its derivative is nowhere vanishing (so conformal) but $\exp 0 = \exp 2 \pi i$ (so not injective).

Proposition. The derivative of an injective holomorphic function $f : D \to \mathbb{C}$, where $D$ is an open subset in $\mathbb{C}$, is nowhere vanishing. In particular, $f$ is conformal.

Proof. We prove the a stronger statement, which will imply the contrapositive of the above proposition. Let $z_0 \in D$. By a change of coordinates if necessary, we may assume that $f(z_0) = 0$ and $z_0 = 0$ without loss of generality. Then, there is a positive integer $n$ and a holomorphic function $h : D \to \mathbb{C}$ such that $f(z) = z^n h(z)$ for all $z$ in $D$ and $h(0) \ne 0$. So in particular there is an open neighbourhood $U$ such that $0 \in U \subseteq D$ and $h$ is nowhere vanishing on $U$. That implies that there is a holomorphic function $g : U \to \mathbb{C}$ such that $g(z)^n = h(z)$ for all $z$ in $U$. (Take a holomorphic branch of the map $w \mapsto w^{1/n}$.) So $f(z) = (z \, g(z))^n$ for all $z$ in $U$.

Note that $z \mapsto z \, g(z)$ has non-zero derivative at $0$ (since $g(0) \ne 0$), so by the inverse function theorem, if $U$ is small enough, $z \mapsto z \, g(z)$ is an invertible holomorphic function on $U$. This implies $f$ is an $n$-to-$1$ function on $U \setminus \{ 0 \}$. But if $f'(0) = 0$, by considering the power series of $f$, we must have $n \ge 2$.

We conclude that a holomorphic function $f : D \to \mathbb{C}$ has derivative vanishing at $z_0$ if and only if it fails to be locally injective at $z_0$.

(This is probably my favourite result in complex analysis.)

Answer 2

It is not possible for a holomorphic function to be injective but not conformal; for a holomorphic f to be injective, its derivative must be non-zero, and conformality of f is precisely equivalent to having $f'(z)\neq 0$.

On the other hand, it is possible for f to be conformal but not injective: $expz:=e^z$ is one example.Conformality follows from the fact that f(z) is analytic , and $\frac{de^z}{dz}=e^z$, and $e^z\neq 0$; (use the identity $e^ze^{-z}=1$ to show $e^z\neq 0$)

EDIT: Conformality follows from the Cauchy-Riemann equations , which show that the effect of f'(z) is that of stretching and rotating. If f'(z) is not zero, then any tangent vector forming an angle $\theta$ will be rotated, under f'(z) by an agle $\theta$, so that two vectors $v,w$ forming an angle $\gamma$ between them, will each be rotated by the same amount, with the effect that the net difference in the angles of the vectors v,w, and their respective images f(v),f(w) is $\theta -\theta=0$