§2.4 The Conformality of Holomorphic Mappings

Let $𝑓:𝑈\to ℂ$ be a holomorphic function defined on an open and connected subset $𝑈\subseteq ℂ$, and let ${𝑧}_0\in 𝑈$ be a point such that ${𝑓}^{\prime }\left({𝑧}_0\right)\ne 0$. Consider a differentiable curve $𝛾\in {𝐶}^1\left(\left[0,1\right]\right)$ with $𝛾\left(0\right)={𝑧}_0$. The direction of the curve at ${𝑧}_0$ is given by the argument of its derivative, namely $Arg\left({𝛾}^{\prime }\left(0\right)\right)$.

The image of $𝛾$ under $𝑓$, defined by $𝜎\left(𝑡\right)=𝑓\left(𝛾\left(𝑡\right)\right)$, is also a smooth curve passing through ${𝑤}_0=𝑓\left({𝑧}_0\right)$. By the chain rule, the derivative of $𝜎$ at $𝑡=0$ is given by

and hence

It follows that

This shows that the change in direction between a curve and its image under $𝑓$ is independent of the curve itself, depending only on the value of ${𝑓}^{\prime }\left({𝑧}_0\right)$.

Now consider two smooth curves ${𝛾}_1,{𝛾}_2\in {𝐶}^1\left(\left[0,1\right]\right)$ such that ${𝛾}_1\left(0\right)={𝛾}_2\left(0\right)={𝑧}_0$, with respective images ${𝜎}_1\left(𝑡\right)=𝑓\left({𝛾}_1\left(𝑡\right)\right)$ and ${𝜎}_2\left(𝑡\right)=𝑓\left({𝛾}_2\left(𝑡\right)\right)$. Then

and by rearrangement,

This equality demonstrates that the angle between two smooth curves at ${𝑧}_0$ is preserved under $𝑓$, provided ${𝑓}^{\prime }\left({𝑧}_0\right)\ne 0$. In other words, holomorphic functions with non-vanishing derivatives preserve angles and orientation locally, a property known as conformality.

Furthermore, for any smooth curve $𝛾\in {𝐶}^1\left(\left[0,1\right]\right)$ passing through ${𝑧}_0$, the limit

shows that infinitesimal lengths are locally scaled by a factor of $\left|{𝑓}^{\prime }\left({𝑧}_0\right)\right|$ under the mapping $𝑓$.

Therefore, at points where $\left|{𝑓}^{\prime }\left(𝑧\right)\right|\ne 0$, the function $𝑓$ is conformal; it preserves angles but not necessarily lengths.