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 = az = ez ln a.

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

a = e u = ex(cosy + isiny) a = -e u = e-x(cosy - isiny);(because ofcos(-y) = cos(y);sin(-y) = -sin(y)) in general u = az = (eln a)z = eln a(x+iy) = ex ln aeiy ln a = exln a(cosylna + isinylna)

Thus the choice of a base e can be compensated via a coordinate transformation: x = xlna;y = ylna

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


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Figure 7.5: Conformal mapping with the complex exponential function u = ez; 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. 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:

z = eiϕ u = eeiϕ = ecos ϕ+i sin ϕ.

We however have cos < 1 in the range π2 < ϕ < 3π2, such that we have ecos 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 2πi at its original position. A strip of the z-plane that is parallel to the real axis of width 2π 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 2πi.


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Figure 7.6: Conformal mapping with the complex exponential function u = ez; Mapping of a circular array with radius 1 and a point grid that has been shifted by 2π 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