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Mathematics LibreTexts

1.8: Complex Functions as Mappings

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A complex function w=f(z) is hard to graph because it takes 4 dimensions: 2 for z and 2 for w. So, to visualize them we will think of complex functions as mappings. That is we will think of w=f(z) as taking a point in the complex z-plane and mapping (sending) it to a point in the complex w-plane.

We will use the following terms and symbols to discuss mappings.

  • A function w=f(z) will also be called a mapping of z to w.
  • Alternatively we will write zw or zf(z). This is read as "z maps to w".
  • We will say that "w is the image of z under the mapping" or more simply "w is the image of z".
  • If we have a set of points in the z-plane we will talk of the image of that set under the mapping.
    For example, under the mapping ziz the image of the imaginary z-axis is the real w-axis.

屏幕快照 2020-08-14 下午8.29.54.png
The image of the imaginary axis under ziz

Example 1.8.1

The mapping w=z2. We visualize this by putting the z-plane on the left and the w-plane on the right. We then draw various curves and regions in the z-plane and the corresponding image under z2 in the w-plane.

In the first figure we show that rays from the origin are mapped by z2 to rays from the origin. We see that

  1. The ray L2 at π/4 radians is mapped to the ray f(L2) at π/2 radians.
  2. The rays L2 and L6 are both mapped to the same ray. This is true for each pair of diametrically opposed rays.
  3. A ray at angle θ is mapped to the ray at angle 2θ.

屏幕快照 2020-08-14 下午8.34.34.png
f(z)=z2 maps rays from the origin to rays from the origin.

The next figure gives another view of the mapping. Here we see vertical stripes in the first quadrant are mapped to parabolic stripes that live in the first and second quadrants.

屏幕快照 2020-08-14 下午8.37.49.png
z2=(x2y2)+i2xy maps vertical lines to left facing parabolas.

The next figure is similar to the previous one, except in this figure we look at vertical stripes in both the first and second quadrants. We see that they map to parabolic stripes that live in all four quadrants.

屏幕快照 2020-08-14 下午8.42.09.png
f(z)=z2 maps the first two quadrants to the entire plane.

The next figure shows the mapping of stripes in the first and fourth quadrants. The image map is identical to the previous figure. This is because the fourth quadrant is minus the second quadrant, but z2=(z)2

屏幕快照 2020-08-14 下午8.44.31.png

Vertical stripes in quadrant 4 are mapped identically to vertical stripes in quadrant 2.

屏幕快照 2020-08-14 下午8.46.02.png

Simplified view of the first quadrant being mapped to the first two quadrants.

屏幕快照 2020-08-14 下午8.46.45.png

Simplified view of the first two quadrants being mapped to the entire plane.

Example 1.8.2

The mapping w=ez. Here we present a series of plots showing how the exponential function maps z to w.

屏幕快照 2020-08-14 下午8.51.30.png

Notice that vertical lines are mapped to circles and horizontal lines to rays from the origin.

The next four figures all show essentially the same thing: the exponential function maps horizontal stripes to circular sectors. Any horizontal stripe of width 2π gets mapped to the entire plane minus the origin,

Because the plane minus the origin comes up frequently we give it a name:

Definition: punctured plane

The punctured plane is the complex plane minus the origin. In symbols we can write it as C - {0} or C/{0}.

屏幕快照 2020-08-14 下午8.54.40.png

The horizontal strip 0y<2π is mapped to the punctured plane

屏幕快照 2020-08-14 下午8.55.27.png

The horizontal strip π<yπ is mapped to the punctured plane

屏幕快照 2020-08-14 下午8.56.13.png

Simplified view showing ez maps the horizontal stripe 0y<2π to the punctured plane.

屏幕快照 2020-08-14 下午8.58.57.png

Simplified view showing ez maps the horizontal stripe π<yπ to the punctured plane.


This page titled 1.8: Complex Functions as Mappings is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.

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