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

Comments on the book Complex Analysis

Here are some corrections and amplifications—addressed primarily to students—for the book Complex Analysis by Theodore W. Gamelin. This list (last modified on ) supplements the author’s errata list.

Page xvii
The word “varifold” in the second line of the Introduction does not exist in the mainstream English language. The author probably means either “manifold” or “variform” (both of which convey the meaning of “having various forms”). Nonetheless, mathematical English does recognize the word “varifold,” but as a noun rather than an adjective. In the subject known as geometric measure theory, a varifold is a sort of generalized surface.
Page 8
In the third displayed equation, an exponent is missing on one of the terms. On the right-hand side of the equation, the term \(-3\cos\theta \sin\theta\) should be \(-3\cos\theta\sin^2\theta\).
Page 13
In the first line of Exercise 1, the words “spherical projection” mean “stereographic projection.” The same comment applies to Exercise 5 on page 14.
Page 20
At line 4, for \(\sin x\) read \(\sin y\).
Page 21
In the first displayed formula in Section 6, the definition of the logarithm function apparently is circular, since \(\log\) is defined in terms of itself. The author is using an unstated convention that when the symbol \(\log\) is followed by a positive real number, the meaning is the real natural logarithm function. Thus \(\log\left|z \right|\) means \(\ln\left|z\right|\).
Page 38
The expression \(\nabla h\) appearing in the statement of the theorem means the gradient vector, that is, the vector of real partial derivatives of \(h\).
Page 45
At line 2, the parentheses are unmatched. The expression should be \((f(w)-f(w_0))/(w-w_0)\).
Page 46
In the first line of Section 3, the second instance of the symbol \(D\) is in the wrong font.
Page 57
In the second line of Exercise 3a, for “all partial derivative” read “all partial derivatives.”
Page 58
In the statement of the theorem, the undefined word “smooth” means “having as many derivatives as is necessary for whatever is being asserted to be true” (as the author explains later, on page 71). You should be aware that many authors assume a “smooth curve” to have the additional property that the derivative \(\gamma'(t)\) is everywhere different from \(0\), which means that the curve has a well-defined tangent vector at each point.
The theorem has a hypothesis that the function is “analytic at a point.” Since analytic functions live on open sets, this expression is to be understood as an abbreviation for “analytic in a neighborhood of the point,” that is, analytic on some disk centered at the point. Actually, the proof uses only that the function has a derivative at the single point \(z_0\).
Page 59
The author defines a conformal mapping to be an angle-preserving mapping that is additionally one-to-one. You should be aware that some authors omit the assumption that the mapping is one-to-one.
Page 63
You should be aware that “fractional linear transformations” are widely known as “linear fractional transformations.” The latter terminology is illogical but standard.
In the first example, the dilation \(z\mapsto az\) might be called a “complex dilation.” If \(a=\rho e^{i\theta}\), then multiplication by \(a\) corresponds to a real dilation by factor \(\rho\) composed with a rotation by angle \(\theta\).
Three lines from the bottom of the page, there are unmatched parentheses. The expression should be \( (az+b)/(cz+d)\).
Page 75
In the calculation proposed in Exercise 7, there is an implicit assumption that the change of variables preserves orientation, that is, the Jacobian matrix \(J_F\) has positive determinant. If the change of variables reverses orientation, then both stated equations need a minus sign on one side.
Pages 81–82
The author speaks a bit casually of a differential being independent of path when he means that the integral of the differential is independent of path.
Page 139
At line 5, what the author means by “anything can happen” is that the series might converge everywhere on the boundary circle or diverge everywhere on the boundary circle or converge at some boundary points and diverge at other boundary points.
Page 151, Exercise 5
Although Exercise 4 defines \(f(w)\) only when \(w\in\C\setminus E\), the integral makes sense for every complex number \(w\) when \(E\) is the unit disk: when \(w\) is inside the disk, the integrand has a singularity, but the singularity is integrable.
Page 154, Exercise 4
The author’s definition of Bernoulli numbers differs from the most common convention. The author’s \(B_k\) corresponds to what most people would write as \( (-1)^{k-1}B_{2k}\).
Page 159, Lemma
In the special case that \(R(z_1)=\infty\), the radius \(R(z_2)\) will be \(\infty\) too. The left-hand side of inequality (8.1) then becomes the indeterminate expression \(\infty-\infty\), which here needs to be interpreted as \(0\) to make the statement of the lemma true.
Page 165, Section 1, first paragraph
The wording “the sum of a function analytic inside the annulus and a function analytic outside the annulus” is confusing. As the statement of the theorem makes clear, the meaning is “the sum of a function analytic inside the outer boundary circle of the annulus and a function analytic outside the inner boundary circle of the annulus.”
Page 185
Notice that the theorem solves Exercise 3 on page 119 in section IV.5.
Page 196, Residue Theorem
The hypothesis that there are only finitely many singularities in the domain \(D\) is actually unnecessary. If there were infinitely many singularities in \(D\), all isolated, then the singularities would have an accumulation point on the boundary, so \(f\) would fail to be analytic in a neighborhood of this boundary point, contrary to hypothesis.
Page 198, Exercise 1(i)
The letters \(k\) and \(n\) are supposed to represent arbitrary natural numbers.
Page 200, Example
Notice that what is computed is the symmetric limit \(\lim_{R\to\infty} \int_{-R}^R\), but the doubly improper integral \(\int_{-\infty}^\infty\) in general means two independent limits \(\lim_{R\to\infty} \lim_{S\to\infty} \int_{-R}^S\). In the case at hand, however, the original real integrand is an even function, so computing a single limit suffices.
Page 210
The “principal branch of \(\log z\)” is not the function that the author denotes with a capital letter \(\Log z\), for he needs here a function that is analytic at points of the negative part of the real axis. You could use a branch of the logarithm that has a branch cut on the negative part of the imaginary axis.
Page 213, Example
The author actually computes \(\lim_{\varepsilon\to 0} \lim_{R\to\infty} \left( \int_{-R}^{1-\varepsilon} + \int_{1+\varepsilon}^R\right)\), apparently a principal value at both \(1\) and \(\infty\). Since the integrand is absolutely integrable at infinity, however, the principal value at infinity equals the ordinary improper integral.
Page 220
In the line after equation (8.2), the symbol \(\C_{\varepsilon}\) should be \(C_{\varepsilon}\).