Skip to main content
Logo image

A First Course in Complex Analysis

Matthias Beck, Gerald Marchesi, Dennis Pixton, Lucas Sabalka

Exercises 1.5 Exercises

2.

Find the real and imaginary parts of each of the following:
🔗

(a)

\(\frac{z - a}{z + a}\) for any \(a \in \R\)
🔗
🔗

(b)

\(\frac{3 + 5i}{7i + 1}\)
🔗
Answer.
\(\frac {19} {25} - \frac {8} {25} i\)
🔗
🔗
🔗

(c)

\(\left( \frac{-1 + i\sqrt{3}}{2} \right)^3\)
🔗
Answer.
\(-1\)
🔗
🔗
🔗

(d)

\(i^n\) for any \(n \in \Z\)
🔗
🔗
🔗

3.

Find the absolute value and conjugate of each of the following:
🔗

(b)

\((2 + i)(4 + 3i)\)
🔗
Answer.
\(5 \sqrt{5}\text{,}\) \(5-10i\)
🔗
🔗
🔗

(c)

\(\frac{3 - i}{\sqrt{2} + 3i}\)
🔗
Answer.
\(\sqrt{\frac{10}{11}}\text{,}\) \(\frac{3 + i}{\sqrt{2} - 3i}\)
🔗
🔗
🔗
🔗

6.

Write in both polar and rectangular form:
🔗

(b)

\(\frac{d}{\diff\phi} \, e^{\phi + i\phi}\)
🔗
🔗
🔗

7.

Show that the quadratic formula works. That is, for \(a,b,c \in \R\) with \(a \ne 0\text{,}\) prove that the roots of the equation \(az^2+bz+c = 0\) are
\begin{equation*} \frac{-b\pm \sqrt{b^2-4ac}}{2a} \,\text{.} \end{equation*}
🔗
Here we define \(\sqrt{b^2-4ac} = i \sqrt{-b^2+4ac}\) if the discriminant \(b^2-4ac\) is negative.
🔗
🔗

9.

Find all solutions of the equation \(z^2 + 2 z + (1 - i) = 0\text{.}\)
🔗
Answer.
\(\pm \, e^{i \frac \pi 4} - 1\)
🔗
🔗
🔗

10.

Fix \(a \in \C\) and \(b \in \R\text{.}\) Show that the equation \(|z^2| + \Re(az) + b = 0\) has a solution if and only if \(|a^2| \geq 4b\text{.}\) When solutions exist, show the solution set is a circle.
🔗
🔗

11.

Find all solutions to the following equations:
🔗

(a)

\(z^6 = 1\)
🔗
Answer.
\(z = e^{i \frac \pi 3 k} , \ k=0,1,\dots,5\)
🔗
🔗
🔗

(b)

\(z^4 = -16\)
🔗
Answer.
\(z = 2 \, e^{i \frac \pi 4 + \frac \pi 2 k} , \ k=0,1,2,3\)
🔗
🔗
🔗
🔗

12.

Show that \(|z| = 1\) if and only if \(\frac 1 z = \overline{z}\text{.}\)
🔗
🔗

13.

Show that
🔗

(a)

\(z\) is a real number if and only if \(z = \o z\text{.}\)
🔗
🔗

(b)

\(z\) is either real or purely imaginary if and only if \((\o{z})^2 = z^2\text{.}\)
🔗
🔗
🔗

15.

Show that if \(z_1 \, z_2 = 0\) then \(z_1 = 0\) or \(z_2 = 0\text{.}\)
🔗
🔗

17.

Fix a positive integer \(n\text{.}\) Prove that the solutions to the equation \(z^n = 1\) are precisely \(z = e^{ 2 \pi i \frac m n }\) where \(m \in \Z\text{.}\)
🔗
Hint.
To show that every solution of \(z^n = 1\) is of this form, first prove that it must be of the form \(z = e^{ 2 \pi i \frac a n }\) for some \(a \in \R\text{,}\) then write \(a = m + b\) for some integer \(m\) and some real number \(0 \le b \lt 1\text{,}\) and then argue that \(b\) has to be zero.
🔗
🔗
🔗

18.

Show that
\begin{equation*} z^5 - 1 \ = \ \left( z-1 \right) \left( z^2 + 2z \cos \tfrac{ \pi }{ 5 } + 1 \right) \left( z^2 - 2z \cos \tfrac{ 2 \pi }{ 5 } + 1 \right) \end{equation*}
and deduce from this closed formulas for \(\cos \frac{ \pi }{ 5 }\) and \(\cos \frac{ 2 \pi }{ 5 }\text{.}\)
🔗
Answer.
\(\cos \frac{ \pi }{ 5 } = \frac 1 4 (\sqrt 5 + 1)\) and \(\cos \frac{ 2 \pi }{ 5 } = \frac 1 4 (\sqrt 5 - 1)\)
🔗
🔗
🔗

19.

Fix a positive integer \(n\) and a complex number \(w\text{.}\) Find all solutions to \(z^n = w\text{.}\)
🔗
Hint.
Write \(w\) in terms of polar coordinates.
🔗
🔗
🔗

20.

Use Proposition 1.2.7 to derive the triple angle formulas:
🔗

(a)

\(\cos (3 \phi) = \cos^3 \phi - 3 \cos \phi \sin^2 \phi\)
🔗
🔗

(b)

\(\sin (3 \phi) = 3 \cos^2 \phi \sin \phi - \sin^3 \phi\)
🔗
🔗
🔗

21.

Given \(x, y \in \R\text{,}\) define the matrix \(M(x,y) := \displaystyle\begin{bmatrix}x \amp -y \\ y \amp x \end{bmatrix}\text{.}\) Show that
\begin{equation*} M(x,y) + M(a,b) \ = \ M(x+a, \, y+b) \end{equation*}
and
\begin{equation*} M(x,y) \, M(a,b) \ = \ M(xa-yb, \, xb+ya)\text{.} \end{equation*}
🔗
(This means that the set \(\{ M(x,y) : \, x, y \in \R \}\text{,}\) equipped with the usual addition and multiplication of matrices, behaves exactly like \(\C = \{ (x,y) : \, x,y \in \R \}\text{.}\))
🔗
🔗

23.

Sketch the following sets in the complex plane:
🔗

(a)

\(\left\{ z \in \C : \, \left| z - 1 + i \right| = 2 \right\}\)
🔗
🔗

(b)

\(\left\{ z \in \C : \, \left| z - 1 + i \right| \leq 2 \right\}\)
🔗
🔗

(c)

\(\left\{ z \in \C : \, \Re (z+2-2i) = 3 \right\}\)
🔗
🔗

(d)

\(\left\{ z \in \C : \, \left| z - i \right| + \left| z + i \right| = 3 \right\}\)
🔗
🔗

(e)

\(\left\{ z \in \C : \, |z| = |z + 1| \right \}\)
🔗
🔗

(f)

\(\left\{ z \in \C : \, |z-1| = 2 \, |z + 1| \right \}\)
🔗
🔗

(g)

\(\left\{ z \in \C : \, \Re(z^2) = 1 \right \}\)
🔗
🔗

(h)

\(\left\{ z \in \C : \, \Im(z^2) = 1 \right \}\)
🔗
🔗
🔗

24.

Suppose \(p\) is a polynomial with real coefficients.
🔗

(a)

Prove that \(\o{p(z)} = p \left( \o z \right)\text{.}\)
🔗
🔗

(b)

Prove that \(p(z) = 0\) if and only if \(p \left( \o z \right) = 0\text{.}\)
🔗
🔗
🔗

25.

Prove the reverse triangle inequality (Corollary 1.3.5(b)) \(\left| z_1 - z_2 \right| \geq \left| z_1 \right| - \left| z_2 \right| \text{.}\)
🔗
🔗

26.

Use Exercise 1.5.25 to show that
\begin{equation*} \left| \frac{1}{z^2 - 1} \right| \leq \frac{1}{3} \end{equation*}
for every \(z\) on the circle \(C[0,2]\text{.}\)
🔗
🔗

27.

Sketch the sets defined by the following constraints and determine whether they are open, closed, or neither; bounded; connected.
🔗

(e)

\(\abs{z-1}+\abs{z+1}\lt 3\)
🔗
🔗
🔗

29.

Let \(G\) be the set of points \(z \in \C\) satisfying either \(z\) is real and \(-2\lt z\lt -1\text{,}\) or \(|z|\lt 1\text{,}\) or \(z=1\) or \(z=2\text{.}\)
🔗

(a)

Sketch the set \(G\text{,}\) being careful to indicate exactly the points that are in \(G\text{.}\)
🔗
🔗

(b)

Determine the interior points of \(G\text{.}\)
🔗
🔗

(c)

Determine the boundary points of \(G\text{.}\)
🔗
🔗

(d)

Determine the isolated points of \(G\text{.}\)
🔗
🔗
🔗

30.

The set \(G\) in Exercise 1.5.29 can be written in three different ways as the union of two disjoint nonempty separated subsets. Describe them, and in each case say briefly why the subsets are separated.
🔗
🔗

31.

Show that the union of two regions with nonempty intersection is itself a region.
🔗
🔗

32.

Show that if \(A \subseteq B\) and \(B\) is closed, then \(\partial A \subseteq B\text{.}\) Similarly, if \(A \subseteq B\) and \(A\) is open, show that \(A\) is contained in the interior of \(B\text{.}\)
🔗
🔗

33.

Find a parametrization for each of the following paths:
🔗

(a)

the circle \(C[1+i, 1]\text{,}\) oriented counter-clockwise
🔗
Answer.
For instance: \(\gamma(t) = (1 + i) + e^{it}\text{,}\) \(0 \leq t \leq 2\pi\text{.}\)
🔗
🔗
🔗

(b)

the line segment from \(-1-i\) to \(2i\)
🔗
Answer.
For instance: \(\gamma(t) = (-1 - i)(1 - t) + (2i)t = -1 - i + (1 + 3i)t\text{,}\) \(0 \leq t \leq 1\text{.}\)
🔗
🔗
🔗

(c)

the top half of the circle \(C[0, 34]\text{,}\) oriented clockwise
🔗
Answer.
For instance: \(\gamma(t) = 34 e^{(\pi - t)i} = -34 e^{-it}\text{,}\) \(0 \leq t \leq \pi\)
🔗
🔗
🔗

(d)

the rectangle with vertices \(\pm 1 \pm 2i\text{,}\) oriented counter-clockwise
🔗
🔗

(e)

the ellipse \(\{ z \in \C : \, |z-1| + |z+1| = 4 \}\text{,}\) oriented counter-clockwise
🔗
🔗
🔗

34.

Draw the path parametrized by
\begin{equation*} \gg(t) = \cos(t) \left| \cos(t) \right| + i \sin(t) \left| \sin(t) \right| , \qquad 0 \le t \le 2 \pi \,\text{.} \end{equation*}
🔗
🔗

35.

Let \(G\) be the annulus determined by the inequalities \(2\lt \abs z\lt 3\text{.}\) This is a connected open set. Find the maximum number of horizontal and vertical segments in \(G\) needed to connect two points of \(G\text{.}\)
🔗
🔗
🔗
PrevTopNext