As second example for conformal mappings we show the complex exponential function. We generalize it to an arbitrary base :
With we obtain the normal exponential functions, with we obtain the exponential decay function.
Thus the choice of a base can be compensated via a coordinate transformation:
the power function and is shown in Fig7.5 for the simple exponential function with .
The real point 1 is mapped to the real point . Negative real parts of lead to a mapping into the inside of the unit circle, positive ones to a mapping into the outside of the unit circle (marked in the picture by a circle). This has the following reason:
We however have in the range , such that we have .
The fundamental peculiarity of the complex exponential function is made clear in this simulation: If one moves the point grid along the imaginary axis, it is turned in the image plane without additional distortion around the origin and arrives after a shift by at its original position. A strip of the -plane that is parallel to the real axis of width fills a complete Riemannian sheet in the -plane. This also shows the periodicity of the trigonometric functions. Fig.7.6 shows the case of shifting the simulation of Fig7.5 by .