2.4 The Conformality of Holomorphic Mappings

Let be a holomorphic function defined on an open and connected subset , and let be a point such that . Consider a differentiable curve with . The direction of the curve at is given by the argument of its derivative, namely .

The image of under , defined by , is also a smooth curve passing through . By the chain rule, the derivative of at is given by

and hence

It follows that

This shows that the change in direction between a curve and its image under is independent of the curve itself, depending only on the value of .

Now consider two smooth curves such that , with respective images and . Then

and by rearrangement,

This equality demonstrates that the angle between two smooth curves at is preserved under , provided . In other words, holomorphic functions with non-vanishing derivatives preserve angles and orientation locally, a property known as conformality.

Furthermore, for any smooth curve passing through , the limit

shows that infinitesimal lengths are locally scaled by a factor of under the mapping .

Therefore, at points where , the function is conformal; it preserves angles but not necessarily lengths.