1.
Show that if \(f(z) = \frac{ az+b }{ cz+d }\) is a Möbius transformation then \(f^{-1} (z) = \frac{ dz-b }{
-cz+a }\text{.}\)
Exercises 3.6 Exercises
2.
Complete the picture painted by Proposition 3.1.2 by considering Möbius transformations with \(c=0\text{.}\) That is, show that \(f: \C \to \C\) given by \(f(z) = \frac{ az+b }{ d }\) is a bijection, with \(f^{-1} (z)\) given by the formula in Proposition 3.1.2.
3.
4.
Extend Example 3.1.6 by showing that \(f(z) = \frac{ z-1 }{ iz+i }\) maps any circle passing through \(-1\) to a line.
5.
Prove that any Möbius transformation different from the identity map can have at most two fixed points. (A fixed point of a function \(f\) is a number \(z\) such that \(f(z) = z\text{.}\))
6.
Prove Proposition 3.1.4.
7.
Show that the Möbius transformation \(f(z) = \frac{ 1+z
}{ 1-z }\) maps the unit circle (minus the point \(z=1\)) onto the imaginary axis.
8.
Suppose that \(f\) is holomorphic in the region \(G\) and \(f(G)\) is a subset of the unit circle. Show that \(f\) is constant.
9.
Fix \(a \in \C\) with \(|a| \lt 1\) and consider
\begin{equation*}
f_a(z) \ := \ \frac{ z-a }{ 1 - \conj{a} z } \,\text{.}
\end{equation*}
(a)
Show that \(f_a(z)\) is a Möbius transformation.
(b)
Show that \(f_a^{ -1 } (z) = f_{ -a }(z)\text{.}\)
(c)
10.
Suppose
\begin{equation*}
A=\displaystyle\begin{bmatrix}a\amp b\\c\amp d \end{bmatrix}
\end{equation*}
is a \(2\times2\) matrix of complex numbers whose determinant \(ad-bc\) is nonzero. Then we can define a corresponding Möbius transformation on \(\Chat\) by \(T_A(z) = \frac{az+b}{cz+d}\text{.}\) Show that \(T_A\circ T_B = T_{A\cdot B}\text{,}\) where \(\circ\) denotes composition and \(\cdot\) denotes matrix multiplication.
11.
Show that our definition of \(\Chat\) honors the “finite” limit rules in Proposition 2.1.6, by proving the following, where \(a \in \C\text{:}\)
(a)
If \(\lim_{z\to z_0}f(z)=\infty\) and \(\lim_{z\to z_0}g(z)=a\) then \(\lim_{z\to z_0}
(f(z) + g(z)) = \infty \text{.}\)
(b)
If \(\lim_{z\to z_0}f(z)=\infty\) and \(\lim_{z\to z_0}g(z)=a \neq 0\) then \(\lim_{z\to
z_0} (f(z) \cdot g(z)) = \infty \text{.}\)
(c)
If \(\lim_{z\to z_0}f(z)=\lim_{z\to z_0}g(z)=\infty\) then \(\lim_{z\to z_0} (f(z) \cdot g(z)) =\infty \text{.}\)
(d)
If \(\lim_{z\to z_0}f(z)=\infty\) and \(\lim_{z\to z_0}g(z)=a\) then \(\lim_{z\to z_0}
\frac{ g(z) }{ f(z) } = 0 \text{.}\)
(e)
If \(\lim_{z\to z_0}f(z)=0\) and \(\lim_{z\to z_0}g(z)=a \ne 0\) then \(\lim_{z\to
z_0} \frac{ g(z) }{ f(z) } = \infty \text{.}\)
12.
Fix \(c_0, c_1, \dots, c_{d-1} \in \C\text{.}\) Prove that
\begin{equation*}
\lim_{ z \to \infty } 1 + \frac{ c_{d-1} }{ z } + \frac{
c_{d-2} }{ z^2 } + \dots + \frac{ c_0 }{ z^d } \ = \ 1 \, \text{.}
\end{equation*}
13.
Let \(f(z)=\frac{2z}{z+2}\text{.}\) Draw two graphs, one showing the following six sets in the \(z\)-plane and the other showing their images in the \(w\)-plane. Label the sets. (You should only need to calculate the images of \(0\text{,}\) \(\pm2\text{,}\) \(\pm(1+i)\text{,}\) and \(\infty\text{;}\) remember that Möbius transformations preserve angles.)
(a)
(b)
(c)
(d)
(e)
(f)
14.
Find Möbius transformations satisfying each of the following. Write your answers in standard form, as \(\frac{az+b}{cz+d}\text{.}\)
(a)
\(1\to0, 2\to1, 3\to\infty\)
(b)
\(1\to0, 1+i\to1, 2\to\infty\)
(c)
\(0\to i, 1\to1, \infty\to-i \)
15.
Using the cross ratio, with different choices of \(z_k\text{,}\) find two different Möbius transformations that transform \(C[1+i,1]\) onto the real axis plus \(\infty\text{.}\) In each case, find the image of the center of the circle.
16.
Let \(\gamma\) be the unit circle. Find a Möbius transformation that transforms \(\gamma\) onto \(\gamma\) and transforms \(0\) to \(\frac 1 2\text{.}\)
17.
Describe the image of the region under the transformation:
(a)
(b)
(c)
18.
Find a Möbius transformation that maps the unit disk to \(\{ x+iy \in \C : \, x+y > 0 \}\text{.}\)
19.
20.
Find each Möbius transformation \(f\text{:}\)
(a)
(b)
(c)
\(f\) maps the \(x\)-axis to \(y=x\text{,}\) the \(y\)-axis to \(y=-x\text{,}\) and the unit circle to itself.
21.
(a)
Find a Möbius transformation that maps the unit circle to \(\{ x+iy \in \C : \, x+y = 0 \}\text{.}\)
(b)
Find two Möbius transformations that map the unit disk
\begin{equation*}
\{ z \in \C : \, |z| \lt 1 \} \qquad \text{ to }
\qquad \begin{aligned}\amp \{ x+iy \in \C : \, x+y > 0
\} \ \text{ and } \\ \amp \{ x+iy \in \C : \, x+y \lt
0 \} \, , \end{aligned}
\end{equation*}
respectively.
22.
Given \(a \in \R \setminus \{ 0 \}\text{,}\) show that the image of the line \(y = a\) under inversion is the circle with center \(\frac{-i}{2a}\) and radius \(\frac{1}{2|a|}\text{.}\)
23.
Suppose \(z_1\text{,}\) \(z_2\) and \(z_3\) are distinct points in \(\Chat\text{.}\) Show that \(z\) is on the circle passing through \(z_1\text{,}\) \(z_2\) and \(z_3\) if and only if \([z,z_1,z_2,z_3]\) is real or \(\infty\text{.}\)
24.
Prove that the stereographic projection of Proposition 3.3.3 is a bijection by verifying that \(\phi \circ \phi^{-1}\) and \(\phi^{-1} \circ \phi\) are the identity map.
25.
Find the image of the following points under the stereographic projection \(\phi\text{:}\)
\((0,0,-1), (0,0,1), (1,0,0), (0,1,0)\text{.}\)
26.
Consider the plane \(H\) determined by \(x+y-z = 0\text{.}\) What is a unit normal vector to \(H\text{?}\) Compute the image of \(H \cap \mathbb{S}^2\) under the stereographic projection \(\phi\text{.}\)
27.
Prove that every circle in the extended complex plane \(\Chat\) is the image of some circle in \(\mathbb{S}^2\) under the stereographic projection \(\phi\text{.}\)
28.
Describe the effect of the basic Möbius transformations rotation, real dilation, and translation on the Riemann sphere.
29.
Prove that \(\overline{\sin(z)} = \sin(\overline{z})\) and \(\overline{\cos(z)} = \cos(\overline{z})\text{.}\)
30.
Let \(z=x+iy\) and show that
(a)
\(\sin z = \sin x \cosh y + i \cos x \sinh y\text{.}\)
(b)
\(\cos z = \cos x \cosh y - i \sin x \sinh y\text{.}\)
31.
Prove that the zeros of \(\sin z\) are all real valued. Conclude that they are precisely the integer multiples of \(\pi\text{.}\)
32.
Describe the images of the following sets under the exponential function \(\exp(z)\text{:}\)
(a)
the line segment defined by \(z = iy , \ 0 \leq y \leq
2 \pi\)
(b)
the line segment defined by \(z = 1 + iy , \ 0 \leq y
\leq 2 \pi\)
(c)
the rectangle \(\{ z=x+iy \in \C : \, 0 \leq x \leq 1 ,
\, 0 \leq y \leq 2 \pi \} \text{.}\)
33.
Prove Proposition 3.4.2.
34.
Prove Proposition 3.4.5.
35.
Let \(z=x+iy\) and show that
(a)
\(\abs{\sin z}^2 = \sin^2x+\sinh^2y = \cosh^2y-\cos^2x\)
(b)
\(\abs{\cos z}^2 = \cos^2x+\sinh^2y = \cosh^2y-\sin^2x\)
(c)
If \(\cos x = 0\) then
\begin{equation*}
\abs{\cot z}^2 \ = \ \frac{\cosh^2y-1}{\cosh^2y} \ \le \ 1\,\text{.}
\end{equation*}
(d)
If \(\abs{y}\ge1\) then
\begin{equation*}
\abs{\cot z}^2 \ \le \ \frac{\sinh^2y+1}{\sinh^2y} \ = \
1+\frac1{\sinh^2y} \ \le \ 1+\frac1{\sinh^21} \ \le \
2\, \text{.}
\end{equation*}
36.
Show that \(\tan(iz)=i\tanh(z)\text{.}\)
37.
Draw a picture of the images of vertical lines under the sine function. Do the same for the tangent function.
38.
Determine the image of the strip \(\{z \in \C : - \frac \pi 2 \lt \Re z \lt \frac \pi 2\}\) under the sine function.
Hint.
Exercise 3.6.30 makes it easy to convert parametric equations for horizontal or vertical lines to parametric equations for their images. Note that the equations \(x=A\sin t\) and \(y=B\cos t\) represent an ellipse and the equations \(x=A\cosh t\) and \(y=B\sinh t\) represent a hyperbola. Start by finding the images of the boundary lines of the strip, and then find the images of a few horizontal segments and vertical lines in the strip.
39.
Find the principal values of
(a)
\(\Log(2i)\)
(b)
\((-1)^i\)
(c)
\(\Log(-1+i) \text{.}\)
40.
Convert the following expressions to the form \(x+iy\text{.}\) (Reason carefully.)
(a)
\(e^{i\pi}\)
(b)
\(e^{\pi}\)
(c)
\(i^i\)
(d)
\(e^{\sin(i)}\)
(e)
\(\exp(\Log(3+4i))\)
(f)
\((1+i)^{\frac 1 2}\)
(g)
\(\sqrt{3} \, (1-i)\)
(h)
\(\left( \frac{i+1}{\sqrt{2}} \right)^4\text{.}\)
41.
Is \(\arg(\conj z) = {-\arg(z)}\) true for the multiple-valued argument? What about \(\Arg(\conj z)={-\Arg(z)}\) for the principal branch?
42.
For the multiple-valued logarithm, is there a difference between the set of all values of \(\log( z^2)\) and the set of all values of \(2 \log z\text{?}\)
43.
For each of the following functions, determine all complex numbers for which the function is holomorphic. If you run into a logarithm, use the principal value unless otherwise stated.
(a)
\({ \overline z }^2\)
(b)
\(\frac{ \sin z }{ z^3 + 1 }\)
(c)
\(\fLog ( z - 2i + 1 )\) where \(\fLog (z) = \ln |z| + i \fArg (z)\) with \(0 \leq
\fArg (z) \lt 2 \pi\)
(d)
\(\exp(\o z)\)
(e)
\((z-3)^i\)
(f)
\(i^{z-3} \text{.}\)
44.
Find all solutions to the following equations:
(a)
\(\Log (z) = \frac{ \pi i } 2\)
(b)
\(\Log (z) = \frac{ 3\pi i } 2\)
(c)
\(\exp(z) = \pi i\)
(d)
\(\sin(z) = \cosh(4)\)
(e)
\(\cos(z) = 0\)
(f)
\(\sinh(z) = 0\)
(g)
\(\overline{ \exp(iz) } = \exp( i \, \overline z )\)
(h)
\(z^{\frac 1 2} = 1 + i \text{.}\)
45.
Find the image of the annulus \(1\lt \abs z\lt e\) under the principal value of the logarithm.
46.
Use Exercise 2.5.25 to give an alternative proof that \(\Log\) is holomorphic in \(\C \setminus \R_{ \le 0 } \text{.}\)
47.
Let \(\fLog\) be a branch of the logarithm on \(G\text{,}\) and let \(H := \{ \fLog(z) : \, z \in G \}\text{,}\) the image of \(\fLog\text{.}\) Show that \(\fLog : G \to H\) is a bijection whose inverse map is \(f(z) : H \to G\) given by \(f(z) = \exp(z)\) (i.e., \(f\) is the exponential function restricted to \(H\)).
48.
49.
50.
Prove that \(\exp(b \log a)\) is single valued if and only if \(b\) is an integer. (Note that this means that complex exponentials do not clash with monomials \(z^n\text{,}\) no matter which branch of the logarithm is used.) What can you say if \(b\) is rational?
51.
Describe the image under \(\exp\) of the line with equation \(y=x\text{.}\) To do this you should find an equation (at least parametrically) for the image (you can start with the parametric form \(x=t,\,y=t\)), plot it reasonably carefully, and explain what happens in the limits as \(t\to\infty\) and \(t\to-\infty\text{.}\)
52.
For this problem, \(f(z)=z^2\text{.}\)
(a)
Show that the image under \(f\) of a circle centered at the origin is a circle centered at the origin.
(b)
Show that the image under \(f\) of a ray starting at the origin is a ray starting at the origin.
(c)
Let \(T\) be the figure formed by the horizontal segment from \(0\) to \(2\text{,}\) the circular arc from \(2\) to \(2i\text{,}\) and then the vertical segment from \(2i\) to \(0\text{.}\) Draw \(T\) and \(f(T)\text{.}\)
(d)
53.
As in Exercise 3.6.52, let \(f(z)=z^2\text{.}\) Let \(Q\) be the square with vertices at \(0\text{,}\) \(2\text{,}\) \(2+2i\) and \(2i\text{.}\) Draw \(f(Q)\) and identify the types of image curves corresponding to the segments from \(2\) to \(2+2i\) and from \(2+2i\) to \(2i\text{.}\) They are not parts of either straight lines or circles.
Hint.
You can write the vertical segment parametrically as \(z(t)=2+it\text{.}\) Eliminate the parameter in \(u+iv=f(z(t))\) to get a \((u,v)\) equation for the image curve.) Exercise 3.6.52 and Exercise 3.6.53 are related to the cover picture of the print version of this book.
