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

Comments on the book Complex Made Simple

Here are some corrections and amplifications—addressed primarily to students—for the book Complex Made Simple by David C. Ullrich. This list is disjoint from the author's errata list (dated 8 June 2009).

Page 23, Proof
There is no need to set up the proof as a proof by contradiction. Precisely the same argument shows directly that \(\delta=0\).
Page 25, lines 4–5 of the proof
To say that the complex number \(h\) “is small enough” is an informal way of saying that \(h\) is close enough to \(0\) or, equivalently, that the modulus of \(h\) is small enough. Strictly speaking, to say that a complex number is “small” or “large” is meaningless, for there is no natural order relation on the complex numbers.
Page 45, part (ii) of Proposition 3.14
The density of the range of a function in a punctured neighborhood of an essential singularity is commonly known as the Casorati–Weierstrass theorem.
Page 45, first line of the proof
The letter \(x\) in the sum should be \(z\).
Page 45, Proof
Despite the parenthetical remark at the end of the second paragraph of the proof, Corollary 3.10 is neither necessary nor sufficient to finish the argument. The sticking point is showing that the series \( \sum_{n=-N}^\infty c_n (w-z)^n\) converges in the original punctured disk \(D'(z,r)\), not merely in \(D'(z,\rho)\). An equivalent assertion is that the series \(\sum_{n=0}^\infty a_n (w-z)^n\) converges in \(D(z,r)\), not merely in \(D(z,\rho)\). The latter series equals the function \( (w-z)^N f(w)\) in \(D'(z,\rho)\), so this function of \(w\) has a removable singularity at \(z\). Removing the singularity produces a function that is holomorphic in \(D(z,r)\) (since \(f\) is holomorphic in \(D'(z,r)\)) and so has a power series expansion that converges in \(D(z,r)\). By uniqueness of the coefficients in a power series, this series is identical to \(\sum_{n=0}^\infty a_n (w-z)^n\), which therefore does converge in \(D(z,r)\), as required.
Page 46, paragraph following the Definition, lines 2–3
The first instance of the hyphenated word “holomorphic” has the letters p and r transposed.
Page 47, Exercise 3.4
A stronger conclusion can be deduced: namely, \( |c|\le 1\).
Page 47, Exercise 3.5
A stronger conclusion can be deduced: namely, the degree of the polynomial is at most \(n\).
Page 48, Exercise 3.8
A stronger conclusion can be deduced: namely, \(|c|\le 2\).
This exercise is not merely a made-up problem for practice: the result will be used in Section 6.3.
Page 48, Exercise 3.9
In the displayed formula, the differential \(dx\) should be \(dz\).
Page 48, Exercise 3.11
Notice that the hypothesis is unnecessarily strong: the function \(f\) needs to be holomorphic only in a punctured disk, not in the whole punctured plane. In other words, the problem is essentially local.
The hint offers one approach to solving the exercise. An alternative method is to apply the classification of isolated singularities developed at the end of the chapter. If the indicated function \(f\) did exist, then the real part of \(f\) would tend to \(-\infty\) at the puncture (because the modulus of an exponential is the exponential of the real part). You should be able to show that such behavior of the real part of \(f\) is impossible if \(f\) has a removable singularity; impossible for a different reason if \(f\) has a pole; and impossible for a third reason if \(f\) has an essential singularity.
Pages 48–49, Exercises 3.13 and 3.14
These exercises are versions of the Schwarz Lemma, about which more is coming in Section 8.4.
Page 49, Exercise 3.16
Part of the point of the exercise is that the integral can be evaluated without having to know an antiderivative of the complex exponential function.
Page 49, Exercises 3.17 and 3.18
The unidentified symbol \(c\) is intended to represent a constant.
Page 53, Proof.
In the third sentence, the observation that “any such curve is in fact contained in one of the components of \(V\)” is not relevant. The reason “we may suppose that \(V\) is connected” is simply that if the conclusion (namely, the existence of \(F\) such that \(f=F'\)) holds on each component of \(V\), then the conclusion holds on \(V\). Notice that the hypothesis of the theorem is trivially inherited by each component of \(V\).
In the fourth sentence, the observation “that there is a smooth path joining any two points of \(V\)” is not quite trivial. This assertion is the statement that a connected open subset of the complex numbers is necessarily path connected, and via a smooth curve. See page 471 in Appendix 4.
Page 55, Proof of Proposition 4.2
That (i) implies (ii) is known already from the proof of Proposition 4.1.
Page 55, Proof of Proposition 4.3
The proof assumes that the open set \(V\) is connected. You are supposed to see why there is no loss of generality in this assumption.
Page 56, parenthetical remark after Proposition 4.5
A more compelling reason that \(0\notin V\) is that in the contrary case, the Example at the top of page 20 shows that the integral over a small circle centered at \(0\) would be defined and different from zero.
Page 57, Definition
You are supposed to be able to see why the notion of index is well defined (that is, independent of the choice of the function \(\theta\)).
Page 58, Proof
In lines 9–10 of the proof, the two instances of \(L_1(0)\) should be \(L_1(\gamma(0))\).
In the next line, “for \(j\ge 2\)” should be “for \(1\le j\le n-1\)”.
Page 60, second line from the bottom
The verb “exist” should be “exists” (to agree with the singular subject “a branch”).
Page 64, line 13
The statement that \(g(w,z_n)\to g(w,z)\) is valid but apparently not the statement needed in the proof: namely, that \(g(z_n,w)\to g(z,w)\). The latter statement is valid for precisely the same reason, but also because (did you notice?) the function \(g\) is symmetric in the two variables \(z\) and \(w\).
Page 65, first paragraph
Notice that \(F_2\) is holomorphic not only in \(\Omega\) but also in the larger open set \(\C\setminus \Gamma^*\).
Page 67, Theorem 4.11
As mentioned on page 46, the Laurent series converges not only uniformly on compact sets but even “absolutely and uniformly” on compact subsets of the annulus. This conclusion is implicit in the proof.
Page 79, Theorem 5.0
The hypothesis that “\(f\) is not constant on any component of \(V\)” is unnecessary and should be deleted. The concern is that \(f\) might be equal to the constant \(0\) on some component of \(V\), but you should be able to deduce from the other hypotheses that such a component is disjoint from both \(\gamma^*\) and \(\Omega\), so the proof goes through without essential change. Notice, moreover, that the extra hypothesis is absent from the statement of Rouché's theorem (Theorem 5.4), which is the main application of Theorem 5.0.
Page 80, Lemma 5.2
The lemma can be improved to a biconditional statement: namely, the line segment joining \(z\) and \(w\) avoids the origin if and only if \( |z-w| \lt |z|+|w|\) (with strict inequality). The lemma can be viewed as a characterization of when equality holds in the triangle inequality.
Page 81, Proof of Lemma 5.2
The conclusion is geometrically obvious, for making a rotation reduces to the situation that \(z\) and \(w\) lie on the real axis on opposite sides of the origin. Thus the proof is saying little more than that if \(x\) and \(u\) are positive real numbers, then \( x- (-u)=x+u\).
Page 82, Proof of Theorem 5.4
You should observe that the strict inequality in the hypothesis forces \(f(z)\) and \(g(z)\) to be different from \(0\) for every \(z\) in \(\gamma^*\). This property is a key hypothesis in Theorem 5.0, on which Theorem 5.4 depends.
Page 83, Proof of Theorem 5.5
In the second displayed formula in the proof, the symbol \(S\) should be \(f(S)\).
Page 85, Exercises
At the beginning of the exercises, delete the words “Recall that”, unless you have read ahead to page 130, where the notation \(\mathbb{D}\) seems to appear for the first time in the exposition.
Page 85, Exercise 5.4
In the second line, for “non-negative real axis removed” read “non-positive real axis removed”.
Page 85, Exercise 5.5
Methods of topology can be used to show that the hypothesis of holomorphicity is superfluous. The statement that a continuous self-mapping of a closed ball in Euclidean space of arbitrary dimension necessarily has a fixed point is known as the Brouwer fixed-point theorem.
Page 87, last line
For “this this” read “this”.
Page 90, Proof of Lemma 6.1.1
Explicit bounds are available with no more work. You should be able to show that if \(n\) is an integer and \(\Re (z)=n+\tfrac{1}{2}\), then \(|\cot(\pi z)| \lt 1\), and if \(|\Im( z)|\ge 1\), then \(|\cot(\pi z)| \lt 2\). In other words, the constant \(M\) in the statement of the lemma can be taken equal to \(2\) (for example).
Page 94, Note
The claim “we will try to avoid saying that an infinite product converges or diverges” breaks down in Chapter 13 (for instance Theorems 13.1 and 13.2).
Page 94, Proof of Lemma 6.1.4
In the final sentence of part (i), the reference to Lemma 6.1.4 should be to Lemma 6.1.3.
Page 95, first paragraph
Since differentiation is a local calculation, and a function \(f\) that is zero-free in a small disk does have a logarithm there, the suggested argument to justify that \(L(fg)=L(f)+L(g)\) is not merely “plausible” but indeed rigorous.
Page 95, Proposition 6.1.5
The hypothesis “that \(f\) is not identically zero” is unnecessary. Indeed, if \(f\) is identically equal to zero, then no compact set \(K\) exists with the indicated property, so the statement holds trivially.
Page 96, last sentence of the proof
The last sentence is not needed, for Lemma 6.1.6 already yields that \(P(z)=c\sin(\pi z)\) for all values of \(z\). In other words, the continuity argument is already contained in Lemma 6.1.6.
Page 97, middle of page
The fourth displayed equation has an error corrected on the author's list (the second sum should start with \(n\) equal to \(2\)) but also a second error: the integral should have upper limit \(\infty\), not \(n\).
Page 98, Hint
A useful preliminary trick is to express \(\phi\) as the sum of an even function and an odd function. For the odd piece, both sides of the equation vanish, so the problem reduces to the case of an even function \(\phi\) (in which case \(\phi'\) is odd and \(\phi''\) is even). By symmetry, the right-hand side becomes \(\int_0^{1/2} (t-\frac{1}{2})^2 \phi''(t)\,dt\). This formulation reduces the chance of making a sign blunder in the subsequent calculation.
Page 100, third paragraph
The argument at the end of the paragraph justifies that \(P(z+1)=-P(z)\) when \(z\) is an integer by invoking continuity. This reasoning is valid, but a simpler justification is that both sides of the equation are equal to zero when \(z\) is an integer!
Page 101, line 2
The reference to the nonexistent Theorem 6.1.7 should be to Theorem 6.2.0. The same correction applies at line 16.
Page 106, Proposition 7.2
In the final sentence of the proposition, “for all \(z\in K\)” should be “for all \(z\in V\)”.
The hypothesis that \(f\) is nonconstant is redundant, for another hypothesis states that the derivative of \(f\) is never equal to zero. Indeed, part of the interest in the subsequent Proposition 7.3 is that nonconstancy of \(f\) becomes an essential hypothesis there.
Page 107, Proof of Proposition 7.2
An implicit hypothesis about the sequence \( (\alpha_n)\) is that \(\alpha_n\) is different from \(\alpha\) for every \(n\).
Page 108, line 2
For \(\Ure (f(z)) \ge 0\) read \(\Ure(f'(z))\ge 0\).
Page 108, Proof of Theorem 7.5
There is no need to invoke the chain rule to find the derivative of the inverse function, for the formula is already known from the proof of Proposition 7.2.
Page 110, Theorem 7.7
The hypothesis of “a zero of finite order” simply means that the function is not constant in a neighborhood of the point.
Page 111, Exercise 7.3(i), Hint
Actually, you can even take \(W\) to be \(D(0,1)\), the unit disk.
Page 111, Comment
Corollary 7.8 applies! The problem is local, and making local holomorphic changes of coordinates in the domain and range reduces the problem to the situation that \(W\) is the unit disk and \(f\) is a power function. Accordingly, all that needs to be shown is that if \(m\ge 2\), then there is no continuous \(m\)th root of \(z\) on the unit disk.
Page 113, four lines from the bottom
The point \((z,0)\) does not look like an element of \(\R^3\), but according to the official definition of complex numbers, the symbol \(z\) means an ordered pair \( (x,y)\) of real numbers, so \( (z,0)\) is effectively \( (x,y,0)\).
Page 120, Section 8.2
In the first line, the word “transformations” is missing the second letter r.
Page 121, line 20
The letter \(\mathbb{P}\) stands for “projective”. The set \(\mathbb{P}_1\) is known as one-dimensional complex projective space.
Page 122, line 16
The phrase “for any nonzero \(z\in S\)” is correct but confusing, for the symbol \(z\) here represents a vector in \(\C^2\), whereas three lines above, the symbol \(z\) represents a scalar in \(\C\).
Page 124, line 16
The notation \(\{\,\rho_\theta: \theta\in\R\,\}\) is correct but misleading, since the map \(\theta\mapsto \rho_\theta\) is not injective. The set-builder notation implicitly removes repeated elements.
Page 125, seven lines from the bottom
For “\(\Im(z)=1\)” read “\(\Re(z)=1\)”.
Page 129, line 7
The second component in \(\C_\infty\) is not “the set of all complex numbers of modulus greater than \(1\)” but rather the union of this set and \(\{\infty\}\).
Page 130, top half
The difficulty in interpreting \(\phi_C\) is merely that the description of circles in terms of real parts is less familiar than the description of circles in terms of absolute values. Generalizing Figure 8.1, observe that if complex numbers \(a\) and \(b\) are opposite ends of a diameter of a given circle, then a point \(z\) (different from \(a\) and \(b\)) lies on that circle if and only if the vectors \(z-a\) and \(z-b\) are orthogonal, which is equivalent to saying that \[ \Re \frac{z-a}{z-b} =0. \] This real part evidently is positive when \(|z|\) is large, so continuity considerations show that \(z\) lies inside the circle precisely when \[ \Re \frac{z-a}{z-b} \lt 0. \] Now there is no mystery about why the transformation \[ -i \frac{z-1}{z+1} \] (which equals \(\phi_C\)) maps the unit disk to the upper half-plane. Sometimes more useful than \(\psi_C\) or \(\phi_C\) is the transformation \[ -i \frac{z-i}{z+i}. \] You should be able to show that this transformation is self-inverse and simultaneously maps the unit disk to the upper half-plane and the upper half-plane to the unit disk.
Page 131, Proof of Theorem 8.4.0
The version of the Maximum Modulus Theorem stated in the text (Theorem 3.11) is the nonexistence of a local maximum. What is needed here is a global version of the theorem (which is stated in Exercise 3.12).
Page 132, Proof of Lemma 8.4.2
In the first line of the proof, the “trivial calculation” can be sidestepped by observing that \(\phi_a\) not only interchanges \(0\) with \(a\) but also interchanges \(1/\bar a\) with \(\infty\). Therefore \(\phi_a \circ \phi_a\) fixes the four points \(0\), \(a\), \(1/\bar a\), and \(\infty\) of \(\C_\infty\) and so is the identity transformation (by Exercise 8.5).
Three lines from the bottom of the page, instead of using that \(\phi_a\) is a homeomorphism of \(\C_\infty\), invoke the Maximum Modulus Theorem to deduce that \(\phi_a\) maps \(\mathbb{D}\) into itself. Being self-inverse, the transformation \(\phi_a\) therefore maps \(\mathbb{D}\) bijectively to itself.
Page 138, line 4
For “ordinary triangle” read “ordinary triangle inequality”.
Page 139, Hint
Simpler is to use the definition of \(d\) to write \[ \frac{d(f(z),f(w))} {d(z,w)} = \left| \frac{f(z)-f(w)} {z-w} \right | \, \left| \frac{1-\overline{ z} w} {1- \overline{f(z)} f(w)}\right| \] and then to invoke Lemma 4.8.
Page 139, paragraph following the Hint
In the third line, the absolute-value signs around \(\Phi\) are redundant, for \(\Phi\) takes nonnegative real values by definition.
In the displayed inequality, the restriction that \(z\ne w\) is unneeded, since both sides vanish when \(z=w\). (Of course, the strict inequality in the following line does require that \(z\ne w\).)
Page 139, Exercise 8.10
Notice that \(\mathcal{L}(\gamma)=0\) if and only if \(\gamma\) is a constant curve.
Page 150, Lemma 9.2.0
The notation \(K^\circ\) means the interior of the set \(K\): see page 463 in Appendix 4.
Page 153, Proof of Lemma 9.2.3
In the second line, for “\(K_{j+1}\subset K_j\)” read “\(K_j\subset K_{j+1}\)”.
Page 157, Exercise 9.4
Although the letter \(S\) has denoted a set of functions for the last several pages, now \(S\) is a set of points.
Page 172, paragraph above Exercise 10.1
An alternative argument is that \(\oint f(z)\,dz\) around the unit circle equals \(2\pi i\), not zero, so Cauchy's theorem implies that \(f\) cannot extend to be holomorphic in the unit disk.
Page 178, line 8
For “Mean Value Property” read “the Mean Value Property”.
Page 179, Proof of Theorem 10.1.5
In the first line, for “\(v\colon \overline{\mathbb{D}}:\to \C\)” read “\(v\colon\overline{\mathbb{D}} \to \C\)” (that is, delete the second colon).
In the second line of the displayed formula, the side condition “\(u\in\mathbb{D}\)” should be instead “\(z\in\mathbb{D}\)”.
Page 180, Proof of Theorem 10.1.6
In the second line of the displayed formula, the side condition “\(u\in\mathbb{D}\)” should be instead “\(z\in\mathbb{D}\)”.
Page 180, Proposition 10.1.7
For \(D_3\) read \(D_2\).
Page 188, Exercise 10.9
The letter \(n\) in the parenthetical hint is, of course, not the same as the letter \(n\) in the displayed formula.
Page 189, Exercise 10.12
In the last sentence, the reference to “the previous exercise” means not Exercise 10.11 but rather Exercise 10.10.
Page 192, seven lines from the bottom
For “the comment preceding the proof” read “the comment preceding the statement of the theorem”.
Page 193
At lines 16–17, there is no need to impose the condition that the index of \(\Gamma\) about \(z\) is zero when \(z\) is outside \(D\), for this condition holds automatically when \(D\) is simply connected (by Exercise 4.4).
In the displayed equation at line 20, the statement that \(|\psi|\ge r\) is valid, but the definition of \(K\) implies the stronger property that \(|\psi|\gt r\).
Appendix 2, page 449
The author's notations \(\Ure z\) and \(\Uim z\) are nonstandard ways to typeset \(\Re z\) and \(\Im z\) (namely, the real part of \(z\) and the imaginary part of \(z\)).
The second line of the two-line displayed formula appears visually to be a tautology stating that an expression is equal to itself. The intent is that the complex conjugate of a product is equal to the product of the complex conjugates: namely, \( \overline{(zw)} =\left( \overline{z}\right) \left(\overline{w}\right)\).
Appendix 3, page 457
The parenthetical remark in the middle of the page states that “we will talk about \(a^z\) for other values of \(a\) later.” This promise does not seem to be redeemed.
Appendix 3, page 459
Six lines from the bottom of the page, the arc goes not from \(B\) to \(C\) but rather from the point \( (1,0)\) to \(C\).
Appendix 4, page 464
Exercise A4.14 is commonly known as Cantor's nested-set theorem.
Appendix 4, page 465
Starting with Exercise A4.19 and continuing through this appendix, the symbol \(\rho\) is to be understood as the metric on the space \(Y\), while \(d\) is the metric on the space \(X\).
Appendix 4, Exercise A4.33
Notice that the converse is false: namely, path-connectedness does not imply local path-connectedness. The topologist's comb is a counterexample.