11.6: Examples of conformal maps and excercises

As we’ve seen, once we have flows or harmonic functions on one region, we can use conformal maps to map them to other regions. In this section we will offer a number of conformal maps between various regions. By chaining these together along with scaling, rotating and shifting we can build a large library of conformal maps. Of course there are many many others that we will not touch on.

For convenience, in this section we will let

\[T_0 (z) = \dfrac{z - i}{z + i}. \nonumber \]

This is our standard map of taking the upper half-plane to the unit disk.

Example \(\PageIndex{1}\)

Let \(H_{\alpha}\) be the half-plane above the line

\[y = \tan (\alpha) x, \nonumber \]

i.e., \(\{(x, y) : y > \tan (\alpha) x\}\). Find an FLT from \(H_{\alpha}\) to the unit disk.

Solution

We do this in two steps. First use the rotation

\[T_{-\alpha} (a) = e^{-i \alpha} z \nonumber \]

to map \(H_{\alpha}\) to the upper half-plane. Follow this with the map \(T_0\). So our map is \(T_0 \circ T_{-\alpha} (z)\).

(You supply the picture)

Example \(\PageIndex{2}\)

Let \(A\) be the channel \(0 \le y \le \pi\) in the \(xy\)-plane. Find a conformal map from \(A\) to the upper half-plane.

Solution

The map \(f(z) = e^z\) does the trick. (See the Topic 1 notes!)

(You supply the picture: horizontal lines get mapped to rays from the origin and vertical segments in the channel get mapped to semicircles.)

Example \(\PageIndex{3}\)

Let \(B\) be the upper half of the unit disk. Show that \(T_{0}^{-1}\) maps \(B\) to the second quadrant.

Solution

You supply the argument and figure.

Example \(\PageIndex{4}\)

Let \(B\) be the upper half of the unit disk. Find a conformal map from \(B\) to the upper half-plane.

Solution

The map \(T_{0}^{-1} (z)\) maps \(B\) to the second quadrant. Then multiplying by \(-i\) maps this to the first quadrant. Then squaring maps this to the upper half-plane. In the end we have

\[f(z) = (-i (\dfrac{iz + i}{-z + 1}))^2. \nonumber \]

You supply the sequence of pictures.

Example \(\PageIndex{5}\)

Let \(A\) be the infinite well \(\{(x, y) : x \le 0, 0 \le y \le \pi \}\). Find a confomal map from \(A\) to the upper half-plane.

Figure \(\PageIndex{1}\): Infinite well function. (CC BY-NC; Ümit Kaya)

Solution

The map \(f(z) = e^z\) maps \(A\) to the upper half of the unit disk. Then we can use the map from Example \(\PageIndex{4}\) to map the half-disk to the upper half-plane.

You supply the sequence of pictures.

Example \(\PageIndex{6}\)

Show that the function

\[f(z) = z + 1/z \nonumber \]

maps the region shown in Figure \(\PageIndex{2}\) to the upper half-plane.

Figure \(\PageIndex{2}\): (CC BY-NC; Ümit Kaya)