7.3 Complex Exponential Function

complex-exp

As second example for conformal mappings we show the complex exponential function. We generalize it to an arbitrary base $Math content$:

$Math content$

With $Math content$ we obtain the normal exponential functions, with $Math content$ we obtain the exponential decay function.

$Math content$

Thus the choice of a base $Math content$ can be compensated via a coordinate transformation: $Math content$

the power function and is shown in Fig7.5 for the simple exponential function with $Math content$.

The real point 1 is mapped to the real point $Math content$. Negative real parts $Math content$ of $Math content$ 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:

$Math content$

We however have $Math content$ in the range $Math content$, such that we have $Math content$.

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 $Math content$ at its original position. A strip of the $Math content$-plane that is parallel to the real axis of width $Math content$ fills a complete Riemannian sheet in the $Math content$-plane. This also shows the periodicity of the trigonometric functions. Fig.7.6 shows the case of shifting the simulation of Fig7.5 by $Math content$.