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

$$u={a}^{z}={e}^{zlna}\phantom{\rule{0em}{0ex}}.$$

With $a=e$ we obtain the normal exponential functions, with $a=-e$ we obtain the exponential decay function.

$$\begin{array}{c}a=e\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\to u={e}^{x}\left(cosy+isiny\right)\hfill \\ a=-e\to u={e}^{-x}\left(cosy-isiny\right);\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\left(\text{becauseof}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}cos\left(-y\right)=cos\left(y\right);sin\left(-y\right)=-sin\left(y\right)\right)\hfill \\ \text{ingeneral}u={a}^{z}={\left({e}^{lna}\right)}^{z}={e}^{lna\left(x+iy\right)}={e}^{xlna}\phantom{\rule{0em}{0ex}}{e}^{iylna\cdot}=e{x}^{lna\cdot}\left(cosylna+isinylna\right)\hfill \\ \hfill \end{array}$$

Thus the choice of a base $\ne e$ can be compensated via a coordinate transformation: ${x}^{\prime}=xlna;{y}^{\prime}=ylna$

the power function and is shown in Fig7.5 for the simple exponential function with $a=e$.

The real point 1 is mapped to the real point $e=2.718\dots $. Negative real parts $x<0$ of $z=x+iy$ 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:

$$\begin{array}{c}z={e}^{i\varphi}\to u={e}^{{e}^{i\varphi}}={e}^{cos\varphi +isin\varphi}.\hfill \\ \hfill \end{array}$$

We however have $cos<1$ in the range $\pi \u22152<\varphi <3\pi \u22152$, such that we have ${e}^{cos\varphi}<1$.

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 $2\pi i$ at its original position. A strip of the $z$-plane that is parallel to the real axis of width $2\pi $ fills a complete Riemannian sheet in the $u$-plane. This also shows the periodicity of the trigonometric functions. Fig.7.6 shows the case of shifting the simulation of Fig7.5 by $2\pi i$.