\(\renewcommand{\Re}{\mathop{\textrm{Re}}} \renewcommand{\Im}{\mathop{\textrm{Im}}} \)

Comments on the textbook for Math 407, Spring 2012

Here are some corrections and amplifications to the textbook, Schaum's Outline of Complex Variables, second edition, by Murray R. Spiegel, Seymour Lipschutz, John J. Schiller, and Dennis Spellman, McGraw-Hill, 2009, ISBN 9780071615693.

Section 1.5, formula (2)

The formula is intended to say that the modulus of a quotient equals the quotient of the moduli, but the printed formula has the identical expression on both sides. The formula should read as follows: \[\left| \frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} \qquad \text{if \(z_2\ne 0\).} \]

Problem 1.65

The signs are wrong in the formula, which should say that \(z_1-z_2+z_3-z_4=0\).

Problem 2.66

The universal quantifier is misplaced. The problem should say that for every complex number \(z\), if \(|\sin z|\le 1\), then \(|\Im z| \le \ln(\sqrt{2}+1)\).

Problem 3.8, Solution

At the end of Method 1, notice that the “arbitrary additive constant” is not completely arbitrary: this constant has to be purely imaginary.

Problem 3.44

The statement is incomplete. The derivative does exist at one exceptional point: namely, when \(z=0\).

Problem 3.84

For \(e^{x^2}\) read \(e^{z^2}\).

Problem 3.101

Add the hypothesis that \(f\) is an analytic function.

Problem 4.43

The typesetting is ambiguous. The integral is intended to be \[ \oint_C \frac{dz}{z-2}. \]

Problem 5.33

There is a typographical error in the numerator. For \(\cos\pi2\) read \(\cos\pi z\).

Problem 6.92

The answer shown for part (c) has a typographical error: the initial term \(-1/2\) should be \(-1/z\).

Problem 6.96b

The answer in the book corresponds to the function \(e^{z^2}/z^3\), not the indicated function \(e^z/z^3\).

Problem 7.78

The answer given in the book is \(1/24\), but the correct answer is \(2\pi i/24\), that is, \(\pi i/12\).

Problem 7.47

The answer given in the book is \(-6\pi i\), but the correct answer is \(-6\pi^2 i\).

Problem 8.34b

The answer given in the book is incorrect. The correct equation is \(u^2+v^2=u-v\), which represents a circle with center \((1/2,-1/2)\) and radius \(1/\sqrt{2}\). The circle passes through the point \(0\), and that point is missing from the image (unless the \(z\) plane is taken to be the extended complex plane including the point at infinity).