\( \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \renewcommand{\Re}{\mathop{\textrm{Re}}} \renewcommand{\Im}{\mathop{\textrm{Im}}} \)

Comments on the book Functions of One Complex Variable

Here are some corrections and amplifications—addressed primarily to students—for the book Functions of One Complex Variable, second edition, by John B. Conway. This list supplements the author’s errata list. The corrections on the author’s list have mostly been incorporated into the latest (seventh) printing of the second edition, which is the version you should have if you downloaded an electronic pdf copy from the campus library website.

Page 30
In the line following the definition of \(\limsup\), the mentioned alternative notation of \(\overline{\lim}\) (sometimes used in real analysis) is dubious in the context of complex analysis, where an overline usually denotes complex conjugation.
Page 33
Part (c) of Exercise 6 is a special case of part (a). Possibly the author intended \(k^n\) to be \(n^k\) in part (c), which would make part (c) an independent problem.
Page 35
At the end of the second paragraph and in the third paragraph, the French name Fréchet should have an acute accent on the first vowel. The meaning of “Fréchet differentiable” is that the function admits a best linear approximation. This condition is more restrictive than having a directional derivative in each direction.
Page 36
The formula at lines 7–8 is a bit mysterious, because the limits on the sums are not indicated, and to make the formula valid requires making one of the sums start with index \(-1\). Clearer would be: \( \sum\limits_{n=0}^\infty | a_{n+1} z^n| = \dfrac{1}{|z|} \cdot \sum\limits_{n=1}^\infty | a_n z^n| \lt \infty \).
Page 39
In the statement of Proposition 2.20, the hypothesis that \( g'(z)\ne 0\) is ambiguous, since no explicit quantifier is attached to \(z\). In the preceding sentence, the letter \(z\) denotes a generic point of \(G\), but the function \(g\) is defined on \(\Omega\), not \(G\). So the natural interpretation is that the letter \(z\) is being recycled to denote a generic point of \(\Omega\). In other words, the hypothesis appears to be that \(g'\) is nonzero at every point of \(\Omega\). The proof actually shows, however, that if \(a\) is a specific point of \(G\), and if \(g'\) is known to exist and be nonzero at the single point \(f(a)\), then \(f'(a)\) exists and equals the reciprocal of \(g'(f(a))\). Thus a pointwise hypothesis yields a pointwise conclusion, and a global hypothesis yields a global conclusion.
The hypotheses of the proposition can be weakened. The continuity of \(g\) is a redundant hypothesis for two reasons. First of all, a subsequent hypothesis says that \(g\) is differentiable, and differentiable functions are automatically continuous. Secondly, the function \(f\) is a continuous injection from an open subset of \(\C\) into \(\C\), so Brouwer’s theorem on “invariance of domain” (from topology) implies that \(f\) is automatically a homeomorphism onto its image, which is necessarily an open set. So one might as well assume that \(\Omega=f(G)\), and then the continuity of the inverse function \(g\colon \Omega \to G\) is automatic.
For the concluding statement of the proposition (on page 40), that analyticity of \(g\) implies analyticity of \(f\), the hypothesis that \(g'\) is everywhere different from zero is redundant. Indeed, it will turn out later that the derivative of an injective analytic function is automatically everywhere nonzero. Compare Corollary 7.6 on page 99 in Chapter IV.
Page 40
In the second paragraph following 2.21, the special branch of \(z^b\) defined using the principal branch of the logarithm has a name that is going to be used later (for instance in Exercise 17 at the end of the section); the name is, of course, the “principal branch” of \(z^b\).
In the next paragraph, you should be aware that many authors use the term “domain” instead of “region” to designate a connected open set (usually assumed implicitly to be nonvoid).
Page 44
Corresponding to the above remark about Proposition 2.20 on page 39, the hypothesis in Exercise 10 about \(g'(\omega)\) being nonzero is actually superfluous, but you are not yet in a position to solve the exercise without that hypothesis.
In Exercise 12, notice that (by definition) the number \(0\) is not in the domain of the function \(z^{1/2}\).
In Exercise 14, a stronger result holds by some future developments. It will turn out that the conclusion holds when \(f(z)\) is real merely for all \(z\) in some nonvoid open subset of \(G\).
In Exercise 15, the set is intentionally called \(A\) and not \(A_r\): the answer turns out to be independent of \(r\) (which is part of the interest of the problem).
Page 46
In the paragraph following Theorem 3.4, the requirement that \( \displaystyle \lim_{z\to a} \frac{|f(z)-f(a)|}{|z-a|} \) exist is redundant. A real-differentiable function \(f\) that preserves angles (both magnitude and direction) is conformal.
You should be aware that some authors require conformal maps additionally to be injective.
Page 47
In Definition 3.5, you should be aware that most authors do not distinguish the class of “linear fractional transformations” from the class of “Möbius transformations.” When \(ad-bc\) equals zero, the transformation reduces to a constant function.
Three lines after the definition, “then it follows that” really means “then it is easy to see that”: an easy computation is needed to verify that the composition of two linear fractional transformations is another one.
In the next line, “Hence” is hiding another computation: the side condition that \(ad-bc\ne 0\) is preserved under composition of transformations.
Page 60
In the line following the statement of Theorem 1.4, the name “Stieltjes” is missing a letter.
Page 69
Eight lines from the bottom of the page, the semicolon should be a period.
Page 70
In the first line of Proposition 2.6, insert a space between \(G\) and the parenthetical remark about \(r\) being positive.
Page 76
In Exercise 13, the author’s sign convention for the Bernoulli numbers, \(B_{2n} = (-1)^{n-1} a_{2n}\), is old-fashioned. Most modern authors choose \(B_{2n}\) equal to \(a_{2n}\).
Page 121
Some of the integrals in the Exercises can be done by techniques of first-year real calculus. In particular, 2(b) and 2(h) are much easier to solve by first-year calculus methods than by complex-analysis methods. Exercise 2(f) is simple to solve by either method. Exercises 2(a) and 2(d) are about the same amount of work either way. Exercises 1(a), 1(c), and 1(d) are possibly easier to do via contour integration, depending on how facile you are with first-year real calculus. Exercises 1(b), 2(c), 2(e), and 2(g) cannot be solved via first-year calculus techniques, as far as I know.