\( \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \renewcommand{\Re}{\mathop{\textrm{Re}}} \renewcommand{\Im}{\mathop{\textrm{Im}}} \renewcommand{\Ure}{\mathop{\mathbb{R}\mathrm{e}}} \renewcommand{\Uim}{\mathop{\mathbb{I}\mathrm{m}}} \)

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 156
Three lines from the bottom of the page, the statement that “each function \(\dfrac{1}{f_n}\) is meromorphic” is inexact, because \(f_n\) might be identically equal to zero, in which case the reciprocal is not a meromorphic function. What is true is that if \(n\) is sufficiently large, then \(1\left/f_n\right.\) is meromorphic. Indeed, the hypothesis in this paragraph is that \(f(a)=\infty\), and \(f_n\to f\), so \(f_n(a)\ne 0\) when \(n\) is large enough.
Page 158
The Note following Theorem 3.8 is wrong if \(0\in G\). Indeed, \(f_n(0)=0\) for every \(n\), so the sequence \(\{f_n\}\) certainly does not converge to the constantly infinite function when \(0\in G\). The sequence \(\{f_n\}\) is not normal in any neighborhood of \(0\), for the pointwise limit of \(f_n\) is not continuous at \(0\). The example can be fixed by setting \(f_n(z)\) equal to \(ne^z\) instead of \(nz\).
Page 173
In Exercise 6, the running variable for the infinite product is not indicated. Confusion is possible, since the expression contains both letters \(i\) and \(n\). The intent is that \(i\) represents the imaginary unit, and \(n\) represents the variable running through the natural numbers.