Complex Logarithm

7.5 Complex Logarithm

We finish the chapter on conformal mappings with the natural logarithm. It is well known, that there exists no logarithm for negative numbers in the space of real numbers, since the inverse function $e^{x}$ always leads to a positive number. This limitation is lifted in the space of complex number, in which the logarithm is well defined for all numbers.

Do calculate the complex logarithm one has to use the complex number $z$ in a form, which allows for the separation of real and imaginary part when taking the logarithm. This is not the case for the form $z = x + i y$ , but in polar coordinates it works out:

\begin{matrix} z = r e^{i φ}; r = |z| = \sqrt{x^{2} + y^{2}}; φ = arctan \frac{y}{x}; \\ ln z = ln \sqrt{x^{2} + y^{2}} + i (φ + k 2 π); k integer \\ main value for k = 0 : ln z = ln \sqrt{x^{2} + y^{2}} + i φ = \frac{1}{2} ln (x^{2} + y^{2}) + i φ \end{matrix}

Because of the periodicity of the exponential function wt a period of $2 π i$ , the $z$ plane is mapped identically to an infinitely number of strips parallel to the real axis in the $u$ -plane of width $2 π$ . The main value for $k = 0$ maps the $z$ -plane to $π < y < π$ on the $u$ -plane.

Figure 7.10: Conformal Mapping with the complex log function: Mapping of a point grid and of a circular array around the origin with radius

x

in the

z

-plane to the

u

-plane. A circle with radius

e

in the

z

-plane and a circle with radius

π

in the

u

-plane are drawn in black. The red lines in the

z

-plane mark the boundaries of the main value of the logarithm. Visible shows the transformed curves of parallels to the

x

and

y

axes in the

z

-plane. Play shifts the array parallel to the real axis.

In Fig.7.10 one sees for the quadratic point array the logarithmic compression along the real axis and the compression due to the arc-tangent along the imaginary axis.

For the logarithmic mapping $u = ln z$ one distinguishes 4 regions according to the real part of $z$ in the $z$ -plane

x $\geq$ 1: for these values the logarithm is positive in the space of real numbers. Complex numbers in the this region are transformed to a region with $x > 0$ , which is bounded by the green curve in Fig.7.10. The numbers with equal imaginary part lie on curves orthogonal to the green curve and are marked via the yellow line for $y = 1$ in Fig7.10.
x $\leq$ -1: For these numbers the logarithm does not yield a real solution. Numbers in this region are transformed to regions with $x > 0$ and imaginary parts, that lie on the boundaries of the strip. The bounding curves are analogue but shifted and reflected with respect to the first case. An interesting case is $ln (- 1) = 0 + i π = i π$ , the symmetric solutions are $ln i = \frac{1}{2} i π$ and $ln - i = \frac{3}{2} i π$
0 $<$ x $\leq$ 1: here we have for real numbers real negative values of the logarithm. Numbers in this region are, depending on the imaginary part, transformed into the positive or negative half of the strip. The bounding curves are continuations of the curves for the first case. Complex numbers in this region are transformed to a region with $x > 0$ , which is limited via the green curve in region Fig.7.10. Numbers with identical imaginary parts lie on curves orthogonal to the green curve, that are marked via the yellow line for $y = 1$ in Fig.7.10.
-1 $\leq$ x $<$ 0: Here the logarithm has no real values. Depending on their imaginary part, Numbers in this region are transformed to the negative or positive half plane of the strip and we have $\frac{π}{2} < | y | < π$ for all $y$ . The bounding curves are continuations of the first case.

A circle around the origin is transformed into a line parallel to the imaginary axis, since the real component of the logarithm $0.5 ln (x^{2} + y^{2}) = ln r$ is constant on it. Changing the radius shifts the line in the $x$ -direction.

How are the curves defined, that are shown in Fig.7.10 and appear after activating the switch Visible. In the $z$ -plane $x = 1$ is the boundary for positive logarithms. Therefore the coordinates of the bounding curve in the $u$ -plane are: $x = 0.5 ln (1 + y_{z}^{2}); y = arctan y_{z}$ . For a line with an imaginary part $y = 1$ in the $z$ -plane we get in the $u$ -plane $x = 0.5 ln (x_{z}^{2} + 1); y = arctan (\frac{1}{x_{z}})$ . These two curves are orthogonal to each other.

Further details and hints for experiments are obtained in the description pages of the simulation.

This relatively complex example demonstrates quite clearly the advantage of the interactive simulation over a discussion with formulas and words. When moving the arrays parallel or at a right angle to the imaginary axis the context is immediately grasped visually, which could only be described in lengthy and time consuming verbal descriptions.

From the many examples shown above it should have become clear how to calculate and visualize conformal mappings in general. The examples include on the custom page of the EJS console the code for the other functions that fit on a few lines in an inactive mode. From this it is easy to derive the code for further conformal mappings.

End of Chapter 7