Complex functions map the points of their domain of definition to points within their range in the complex plane (to distinguish these functions from real functions we arbitrarily use capital letters for the function).
important complex functions, like powers, the exponential function and its descendants, among them trigonometric functions and hyperbolic functions, satisfy the property to be holomorphic, which means according to the definition, that they are complex differentiable. This means, that these functions are differentiable in every point of the complex plane and is also independent of the direction along which one approaches the respective point. Such functions can be differentiated an arbitrary number of times and therefore can also expanded into a power series (Taylor series).
Fig.7.1 from the simulation shown in Fig.7.2, that will be described shortly, shows how this looks for the concrete case of the mapping
The mapping with a holomorphic function is conformal that means angle-preserving: curves in the plane intersect under the same angle as the pre-image curves in the -plane. This is initially baffling, since the shapes consisting of the curves are in general distorted via the mapping.
The left window shows the -plane, the right one the -plane. In the -plane a quadratic grid of points, that lay on parallels to the real and imaginary axis, is shown, that is mapped into the -plane, undergoing rotation, stretching (for points outside of the indicated unit circle) or compression (for points inside the unit circle) and resulting in a rhombic shape with curved grid lines. In this case the points on the real axis are transformed to the real axis and therefore the real-valued side of the square stays straight.
On closer examination it becomes evident, that the lines connecting the points in the image plane indeed intersect each other under right angles; the 4 points corresponding to a square of neighboring points in the preimage constitute a square in the image with increasing accuracy for decreasing distance of the points. The conformal angle-preserving property is to be understood in the limit of infinitesimal distances.
The angle-preserving property of conformal mappings is used for practical purposes in engineering, for example to map the solutions of hydrodynamic problems for simple situations to more complex situations. Complex functions are thus not only an abstract mathematical concept, but they have very useful applications.