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Chapter 3: Problem 4

Show that \(w=((1+z) /(1-z))^{2}\) maps the disk \(|z|<1\) onto the plane, excluding the ray \((u, 0), u \leq 0\).

Short Answer

Expert verified
The function \(w=((1+z) /(1-z))^{2}\) transforms the unit disk \(|z|<1\) onto the whole complex plane excluding the ray (0, \(u\)) where \(u\) is less than or equal to 0.

Step by step solution

01

Express the disk in polar coordinates.

By using the polar form for \(z\), we can write \(z = r * e^{i\(\theta\)}\), where \(r\) is the magnitude of \(z\) (with \(0 鈮� r < 1\) since \(|z| < 1\)), and \(\theta\) is the argument of \(z\) (with \(0 鈮� \theta < 2\pi\)).
02

Substitute the polar form of \(z\) into the function.

Substituting the polar form of \(z\) into the function, we get \(w=\(\frac{(1+r * e^{i\(\theta\)})^2}{(1-r * e^{i\(\theta\)})^2}\)\).
03

Simplify the function.

We then simplify this function to its complex form \(u+iv\) by using the binomial theorem and noting that \(e^{2i\(\theta\)} = cos(2\(\theta\)) + i*sin(2\(\theta\)). This results in \(u=(\frac{1 + r^2}{1 - r^2})\) and \(v=(\frac{2r}{1 - r^2}) * sin(2\(\theta\))\).
04

Examine the image of the function.

For the \(u-axis\), as \(r\) approaches 1, \(u\) approaches negative infinity. Hence all negative real numbers are included. On the other hand, when \(r\) approaches 0, \(u\) equals 1. So, all real numbers greater than 0 are also covered. Thus, the image of the function is all real numbers excluding 0. For the \(v-axis\), it has been already shown that \(v\) maps onto the entire set of real numbers. Consequently, we find that the function maps the disk onto the plane, excluding the ray \(u 鈮� 0\).

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Most popular questions from this chapter

If \(z_{1}\) and \(z_{2}\) are distinct fixed points of a bilinear transformation \(w=\) \(T(z)\), show that the transformation may be expressed as $$ \frac{w-z_{1}}{w-z_{2}}=K \frac{z-z_{1}}{z-z_{2}}, $$ where \(K\) is a complex constant.

Let \(w\) be a bilinear transformation from the unit circle onto itself. If \(z_{1}\) is mapped onto \(w_{1}\), show that \(1 / \bar{z}_{1}\) is mapped onto \(1 / \bar{w}_{1}\).

Prove that the linear transformation \(w=a z+b\) maps a circle having radius \(r\) and center \(z_{0}\) onto a circle having radius \(|a| r\) and center \(a z_{0}+b\).

Find conditions for a bilinear transformation to carry a straight line in the \(z\) -plane onto the unit circle \(|w|=1\).

Using the invariance property of the cross-ratio, find a bilinear transformation \(f\) in each of the following cases: (a) \(\\{1, i,-1\\}\) onto \(\\{1,0, i\\}\) (b) \(\\{\infty, i, 0\\}\) onto \(\\{0, i, \infty\\}\) (c) \(\\{-i,-2+i, 3 i\\}\) onto \(\\{4,1+3 i,-2\\}\) (d) \(\\{0,1, \infty\\}\) onto \(\\{-i, 1, i\\}\).
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