Guide to Cultivating Complex Analysis: Changes

May 9th 2024 edition (version 1.6):

Most changes below are quite minor in character, mostly to address issues in clarity, errata, or issues with exercises that I and my students encountered during the last semester teaching with the book. There is essentially nothing new, although a few exercises changed slightly and two new were added, those were all changes to address issues rather than add new material.

1. Make the caption in Figure 1.5 more correct: The sphere is the normal $$S^2$$ sphere.
2. In the example in the last paragraph of 1.3 (page 21), make $$g(z) = c-z$$ so that the limit ends up being c instead of 0.
3. Write Proposition 2.2.5 even for $$n=0$$ but add the special case to the formula then. Avoids a common question. Also use $$\frac{d}{dz}$$ instead of the apostrophe, hopefully that's clearer notation anyway and the one we use in 2.2.4.
4. Very slight change to Exercise 2.2.2 due to the change to Proposition 2.2.5, that is, the student should now also say that $$z^0$$ is holomorphic.
5. On page 36, add a couple of sentences about why we have the modulus squared in the determinant.
6. Add Exercise 2.3.12 proving uniqueness of the coefficients directly without having the more heavyweight formula for the coefficients (which comes a couple of pages later) as the uniqueness fact is much simpler and a good practice, plus useful in another exercise before we get to the formula with the derivatives.
7. On page 54, don't tell the reader to necessarily use sines and cosines, and mark Exercise 3.1.1 as easy.
8. Add assumption that $$g'$$ is continuous to Proposition 3.1.7 as we haven't yet proved it, and add a footnote about it.
9. On page 58, note that the caveat about convergence of paths applies to both the $$dz$$ and the $$|dz|$$ integrals.
10. Move Figure 3.3 up a paragraph, which makes for a much nicer page break and moves Cauchy-Goursat to next page.
11. To be consistent with our definition of triangle, in Proposition 3.2.11, make a note about when the points are collinear, the integral is trivially zero without the hypothesis.
12. Slight change to Exercise 3.2.19: Assume that $$U_1 \cap U_2$$ is nonempty. It is a technicality that makes for a very slightly harder-to-state solution and distracts from the main idea. Plus most students miss this technicality.
13. Make title of subsection 3.3.2 a little more precise so that someone doesn't get the idea that "Morera" is a property of the derivative.
14. Add a few remarks to the proof of 3.4.5, first in the beginning mention that $$f_n \to f$$ uniformly on the circle, second at the end, note explicitly the key fact that the $$d/2$$ circle is a subset of $$K' .$$
15. On page 87, add a comment about Schwarz-Pick providing the bound at other points than 0.
16. After 4.2.2, note that the proof is rather recent and due to Dixon. Also be a little bit more explicit on what this entire function is in the paragraph after the theorem.
17. In the paragraph after Proposition 4.3.6, do not refer to the argument principle section as Exercise 5.4.7 now does not actually fully prove (it was actually quite difficult and didn't really use the argument principle theorem itself) the equivalence of existence of roots and simply connectedness.
18. Use an enumerate in Proposition 4.4.3 for the two parts. As a side benefit it is clear that there is a second part that happens to be on the next page.
19. Make the hint in Exercise 4.4.1 be more useful; pointing the student at subsection 3.4.2 is a better hint.
20. Add footnote to Definition 4.5.9 to note that this is the actual definition of homotopy for any path connected topological space.
21. Change the $$n$$ in Exercise 4.4.6 to $$k,$$ because $$n$$ leads to some hard-to-read proofs unless the students are writing carefully.
22. Reword Exercise 5.1.7 to say that we should prove that the limit of the quotient always exists (if allowed to equal infinity). It is asking for a little bit more, but not only is that a better way to state the exercise, it also makes it simpler to do and less confusing.
23. In Example 5.3.6, show the use of all three propositions for computing the residue.
24. Reword Exercise 5.4.7 to make it clear what to prove, and word it so that the existence of a $$\gamma$$ with winding number 1 around p is given (as that is hard to prove at this point, too hard for an exercise, and this one is designed to teach something different).
25. Add a remark after statement of Lemma 5.6.2 that the image of the disc is a neighborhood of $$f(p).$$
26. Add Exercise 6.3.16 that is at the right place to prove the existence of that cycle that was needed in 5.4.7. Part b would be easy to prove if the student got 5.4.7, but that's OK.
27. Add Remark 6.3.4 about existence of square roots being equivalent to the domain being simply connected as that's an immediate consequence of the proof.
28. Change Exercise 7.2.17 a little by asking to prove the "if and only if." The way it was previously was only asking for the hard part (the if), and so it wasn't giving a good parallel to the theorem. The easy part is actually a good way to start the exercise anyway.
29. On page 180, the last displayed inequality in the proof of Harnack is actually an equality.
30. Simplify Exercise 7.2.26 a little by assuming that $$U$$ is connected to avoid having to think about the technicality of countably many components which is not really important.
31. In Figure 7.6 stop marking $$U_-,$$ which we never defined.
32. Clarify the proof of Rado's theorem.
33. Fix the known errata.

July 19th 2023 edition (version 1.5):

1. Note that the example functions right after definition 6.1.1 are all bounded. Oddly, this changes pagination on the following pages very slightly to be nicer.
2. Consistently use "converges uniformly on compact subsets" instead of sometimes "converges uniformly on compact sets."
3. In the proof of the Riemann mapping theorem, at the end, refer back to the construction of the $$h$$ to make it clear why the maximizer must be onto.
4. Add Exercise 6.3.9 (entire and injective implies onto), a nice application of RMT.
5. In proof of Lemma 6.3.6, use $$m$$ instead of $$n$$ in the start of the proof for the number of discs to avoid overuse of $$n .$$
6. Change the proof of 7.1.10 (Identity) to be more consistent with 7.1.11 (Maximum principle).
7. Fix the known errata.

May 16th 2023 edition (version 1.4):

1. Change "up to multiplicity" to "counting multiplicity." We were using both so stick to one, and the second one is clearer.
2. In the description of the typical application of Hurwitz, be a bit more precise (besides fixing an erratum).
3. In the proof of Proposition B.3.13, use $$j,k$$ for the same thing as in the Definition B.3.12 and Proposition B.3.11 for consistency. This required changing $$k$$ to $$h',$$ which is more consistent with the naming of $$A'$$ anyway. Also use $$p$$ for the fixed point rather than $$x$$ for consistency.
4. A few other minor language/style improvements or clarifications.
5. Fix the known errata.

July 9th 2022 edition (version 1.3):

The main point of this revision was to go through the exercises and shake out as many typos as possible, especially for the exercises that weren't assigned in my class. The changes are all very minor in character.

1. In the proof of Proposition 3.1.8, mark which integral is the arclength integral of the modulus.
2. In Exercise 3.1.13 be more explicit either the path is injective or it is simple closed. The way it was stated made it seem like plausibly one that just bites itself somewhere is also allowed, which was not intended.
3. In the intro to 3.3.2, the example of real differentiable function would only work for bounded $$g$$ so integrate from some $$c \in (a,b).$$
4. In proof of Theorem 3.3.6, maximum modulus principle, consider a closed disc in $$U$$ rather than just saying a circle, it is clearer this way, we are talking about $$|z| \leq r$$ anyhow here.
5. Streamline the wording of Exercise 3.3.19 a bit and include the definition of $$\limsup_{z \to \infty} .$$
6. Change the wording of Theorem 3.3.10 (Liouville) to be more precise.
7. Improve the wording of proof of Theorem 3.3.11 (FTA) a bit more.
8. In the proof of Schwarz's lemma, talk about maximum modulus getting the bound $$|g(z)| \leq \frac{1}{r}$$ instead of $$r|f(z)| \leq |z| .$$ Then only once we show that $$|g(z)| \leq 1$$ go to $$r|f(z)| \leq |z| .$$ Seems more straightforward stated this way.
9. Note that Exercise 3.5.3 is called the Cartan's uniqueness theorem and add it to the index.
10. In Exercise 3.5.7, ask for the condition $$ad-bc > 0$$ rather than just $$ad-bc \not= 0 .$$ While correct that was misleading, and was not what was intended.
11. In proof of Corollary 5.2.3, emphasize where proof of first item ends and the proof of the converse statement starts.
12. In the comments after Definition 5.2.4, emphasize that $$\ell \in \mathbb{Z} .$$
13. Reword Exercise 5.2.23 to be a bit more readable.
14. Reword Exercise 5.4.6 to make it explicit as to what the power sums are.
15. Reword proof of Exercise 5.4.7, it is a bit misleading.
16. Simplify statement of Hurwitz a tiny bit.
17. In Exercise 6.2.2, emphasize that $$U \subset {\mathbb{C}},$$ as the two notions are not the same in an arbitrary metric space.
18. Reword part a) of 6.2.6 to be more explicit.
19. In Theorem 7.4.5 (maximum principle for subharmonic functions), add a footnote to draw attention to the maximum now being a global one.
20. Reword parts b and c of Exercise 9.2.7 to be more logical.
22. Many other minor clarifications and cleanups.
23. Fixed many misspellings, grammar and style issues.
24. Fix the known errata.

May 10th 2022 edition (version 1.2):

The changes are all quite minor, though numerous. The main focus was to weed out any errata, and improve unclear wording. No new content, although there are three new exercises.

1. Add polar form and polar coordinates to the index.
2. On page 18, improve figure 1.4 to include a shaded horizontal and vertical strip in two shades of gray to make it easier to see.
3. Also on page 18, use $$w$$ for the target variable to avoid confusion when defining the annulus and the sector.
4. At end of section 1.3, when discussing arithmetic, note that $$z+\infty=\infty$$ can be defined, it is just that $$\infty+\infty$$ and $$\infty-\infty$$ are undefined.
5. In section 1.4, when defining $$T_a$$ and $$D_a$$ note where $$a$$ lives, in particular, that for $$D_a,$$ the $$a$$ should be nonzero.
6. In subsection 2.1.1 emphasize a tiny bit more the fact that the $$h$$ is complex.
7. In subsection 2.1.1 explicitly state what $$o(|h|)$$ means rather than being vague.
8. Reword the paragraph in front of Proposition 2.1.4 to be clearer.
9. Add the value of $$f'(z_0)$$ in Proposition 2.1.4 in terms of the real partials as we have also proved that above, and it is good to emphasize.
10. At end of subsection 2.1.2 emphasize the exercise for holomorphicity of the exponential.
11. In the first proof of Proposition 2.2.2, handle the $$k=0$$ case explicitly.
12. In second the proof of Proposition 2.2.2, don't refer to $$f$$ and $$g$$ being holomorphic, they are simply complex differentiable at one point.
13. Add some segue sentences in subsection 2.2.1, and rename Proposition 2.2.5 to "Power rule and its consequences."
14. Exercise 2.2.2 should just ask for the power rule, the second item is asked for in Exercise 2.2.3.
15. In Exercise 2.2.11, just use the identically equal sign to make it easier to read.
16. Add a note about what Proposition 2.2.9 means in view of Exercise 1.1.7 after the proof of the proposition.
17. Flip the two paragraphs about mapping properties and the n-to-1 behavior as it makes more sense in this order. This flips the figures 2.2 and 2.3.
18. Proper proof of the Cauchy-Hadamard theorem, it seemed to imply divergence about the series of the absolute values.
19. Add the word "absolutely" to Figure 2.4.
20. Reword Remark 2.3.5. We do definitely need addition, it is multiplication can could plausibly be put off for later.
21. In the proof of 2.4.6, save the "the series converges" only after we proved what it is.
22. Emphasize the idea of factoring out the zero out of a power series after the proof of the identity theorem (2.4.7).
23. In Exercise 2.4.17 emphasize to only show uniqueness.
24. On page 54 when comparing the definitions of line integrals, fix the right hand side and also add an underbrace that shows that the right hand side is precisely our previous definition. Also write $$\gamma(t) = x(t) + i\, y(t)$$ instead of the notation as a point in the plane.
25. After the definition of the line integral, note that the definition still holds even if the derivative would be zero.
26. In simple justification of reparametrization, use $$h' > 0$$ and $$h' < 0$$ instead of increasing decreasing to avoid zero derivative.
27. In Exercise 3.1.5, say $$f\equiv 1$$ instead of $$f=1$$ for emphasis.
28. Fix Proposition 3.1.7 to mention that the new path might not be a path if we are disallowing zero derivatives.
29. In the remark after Definition 3.2.5 emphasize "is" and "is equivalent to" to make it clear what is the difference that we are talking about.
30. In proof of 3.2.9 make sure to state that $$\alpha$$ is a complex number.
31. In proof of Theorem 3.3.11, no need to handle the case when $$P(z)$$ is constant, it is nonconstant by definition.
32. After the proof of 3.3.8, in the example, make it centered at $$p$$ to fix an erratum and mention again that the sup norm is $$M.$$
33. In proof A of 3.4.1, mention that we are applying Cauchy-Goursat (though it could also be one of the other Cauchy theorems).
34. Reword Definition 3.4.4 to read more naturally hopefully.
35. In proof of 3.4.5, one of the equalities was written as an inequality so fix that (it was not really wrong).
36. On page 91, checking that the formula for the principal branch of the logarithm works, note that $$L(1)=0=\operatorname{Log}(1)$$ to start with.
37. On page 95, at the end of the proof of Proposition 4.1.2, it says that $$L_j$$ are branches of $$\log$$ but they are branches of $$\log (z-p).$$
38. In Definition 4.2.1, use "open" rather than "domain", connected is not needed.
39. Improve the wording of proof of Theorem 4.2.2, it was misleading in a couple of bits.
40. Emphasize before Proposition 4.3.7 that simply-connectedness is a topological concept.
41. After statement of Theorem 4.4.2 (Laurent series), expand a bit on the convergence of such a series using what we know about power series.
42. In the introduction to subsection 4.5.1, emphasize that by "path" in this section we mean continuous. Also drop the footnote on definition 4.5.1, I think it was more misleading than helpful.
43. In Proposition 4.5.6 drop the "piecewise-$$C^1$$" it is not really needed any paths will do, though we only really need it for piecewise-$$C^1 .$$
44. Replace the quote at the beginning of chapter 5 with one that irritates me less given what's happening in the world.
45. Tighten up section 5.1 a bit.
46. Add Exercise 5.1.7 to prove L'Hopital's rule.
47. In the paragraph after definition 5.2.1, give parenthetical example of a pole and an essential singularity.
48. In the proof of Theorem 5.2.2, say $$h$$ is holomorphic on $$U$$ rather than just "near $$p$$."
49. Exercise 5.2.8 was worded a bit vaguely, the function is supposed to be "not identically zero".
50. In Corollary 5.2.3, in (i) make the conclusion that $$g$$ has a removable singularity and in (ii) make the hypothesis that it is bounded. that is the way it is meant to go.
51. Exercise 5.2.12, the $$f$$ should be "not identically zero."
52. In the exercises in section 5.3, emphasize that residue theorem should be used for the computations (we're not interested in exercising other calculus tricks here).
53. Rewrite the proof of Rouché (5.4.2) to not replace $$g$$ with $$-g$$ which I forgot about (erratum). The inequality is slightly less appealing, but on the other hand you use the principal branch of log.
54. In Rouché, the winding number should be 0 or 1 for all $$z \notin \Gamma,$$ rather than $$z \in U .$$
55. In section 5.6, add a final note about possibly using inverse function theorem to get the holomorphicity. Also add another note about using the argument principle to prove Lemma 5.6.2.
56. Shortening/tightening some wording to improve the flow improves also the pagination a tiny bit in sections 6.2 and 6.3
57. In the proof of Montel (Theorem 6.2.2), it is slightly easier to just apply fundamental theorem of calculus more directly to the difference $$f(z)-f(p).$$
58. Add Exercise 6.2.10, which is a nice application of Montel to characterizing the radius of convergence.
59. Simplify the statement of Riemann mapping (Theorem 6.3.1). That is, just say both existence and uniqueness at once, instead of following what the proof does.
60. In the proof of Riemann mapping theorem, when showing $$|f'(p)| < |h'(p)|$$ we change $$g(p)$$ to $$\varphi_{-g(p)}(0)$$ and then back again, that is unnecessary. Just leave the $$2g(p)$$ be and change just $$g'(p).$$
61. Further on in the proof of RMT, it says that $$|f_n'(p)|$$ is an increasing sequence which is not right, though we could assume it is. Better to just note that $$0 < |f_n'(p)| \leq |f'(p)|$$ for any $$n.$$
62. Add Exercise 7.3.11 that requires infinitely many reflections.
63. On the bottom of the first page of 9.1, we expand on why the series for $$f$$ cannot converge on the square.
64. In the proof of the reflection principle (10.1.1), mention that $$F$$ is continuous. That is rather immediate, but it should be stated.
65. Before Definition 10.1.2, be even less formal about the introduction of "real-analytic curve" as it seems like we are giving the actual definition but, it's not actually equivalent to 10.1.2.
66. Before Corollary 10.1.3, remove the sentence about "switching sides" it is more confusing than helpful.
67. Corollary 10.1.4 is unnecessarily restrictive, no need for $$\partial\mathbb{D} \subset U$$ if we say $$f(\partial\mathbb{D} \cap U) \subset \partial \mathbb{D}$$ instead of $$f(\partial\mathbb{D}) \subset \partial \mathbb{D}.$$
68. Make the footnote in Definition 10.2.5 into a normal paragraph. The comment is needed a bit later in the monodromy theorem and so shouldn't be relegated to a footnote.
69. Be more precise in the statement of Proposition 10.2.11.
70. In Corollary 10.2.15, only assume that $$V$$ is simply connected. In the proof of that theorem, explicitly note that there is only one inverse as $$U$$ is connected.
71. Flip the order of proof in Proposition A.2.13 to be more logical.
72. In Exercise A.3.11, only ask for the part of the proposition that was not yet proved to avoid confusing matters.
73. Theorem A.5.7 should mention that $$X$$ ought not be empty.
74. A little bit of cleanup of the proof of Theorem B.3.16.
75. Clean up the proof of Proposition B.2.1 to not use sequences as it's simpler without.
76. Some minor clarifications and fixes throughout.
77. Fix the known errata.

December 18th 2020 edition (version 1.1):

Very minor update. Minor wording/grammar improvements. Fix the accents on Rouché and Poincaré.

September 10th 2020 edition (version 1.0):

First version.