Complex Exponential Function

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 $a$ :

u = a^{z} = e^{z ln a} .

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

\begin{matrix} a = e \to u = e^{x} (cos y + i sin y) \\ a = - e \to u = e^{- x} (cos y - i sin y); (because of cos (- y) = cos (y); sin (- y) = - sin (y)) \\ in general u = a^{z} = {(e^{ln a})}^{z} = e^{ln a (x + i y)} = e^{x ln a} e^{i y ln a \cdot} = e x^{ln a \cdot} (cos y ln a + i sin y ln a) \end{matrix}

Thus the choice of a base $\neq e$ can be compensated via a coordinate transformation: $x^{'} = x ln a; y^{'} = y ln a$

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

Figure 7.5: Conformal mapping with the complex exponential function

u = e^{z}

; mapping of a point grid and of a circular array with radius 1 around the origin of the

z

-plane to the

u

-plane. The unit circle is drawn in black. Play shifts the array along the imaginary axis. The parameter

a

can be chosen in the number field a.

The real point 1 is mapped to the real point $e = 2.718 \dots$ . Negative real parts $x < 0$ of $z = x + i y$ 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{matrix} z = e^{i ϕ} \to u = e^{e^{i ϕ}} = e^{cos ϕ + i sin ϕ} . \end{matrix}

We however have $cos < 1$ in the range $π ∕ 2 < ϕ < 3 π ∕ 2$ , such that we have $e^{cos ϕ} < 1$ .

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 $2 π i$ at its original position. A strip of the $z$ -plane that is parallel to the real axis of width $2 π$ 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 $2 π i$ .

Figure 7.6: Conformal mapping with the complex exponential function

u = e^{z}

; Mapping of a circular array with radius 1 and a point grid that has been shifted by

2 π

along the imaginary axis. The picture in the

u

-plane is identical to the picture in Fig.7.5, where the point grid is located at the origin. The unit circle is drawn in black. The boundaries of a period that is symmetric to the origin is marked in red