Guide to Cultivating Complex Analysis: Changes

July 11th 2026 edition (version 1.9):

The main focus of this version is to fix errata and other issues identified with the help of AI (no content was generated by AI). The changes seem numerous, but they are all rather small and there is no new material. Some exercises did change a little due to errata, so those are bolded below as usual.

In Exercise 1.2.2, change lines to rays to fix an erratum.
In definition of the complex exponential remove the \(e^x e^{iy}\) form as it may be confusing and we really give the \(e^{i\theta}\) formula later anyway, no need to confuse things further here.
In Exercise 1.3.5, explicitly say that \(z \in \mathbb{C}\) at the end, rather than leaving it implicit.
In Exercise 1.3.6, change the "whenever" to \(0 < |z-z_0| < \delta\) to fix an erratum.
When defining the relation for the complex projective space, say that \(\lambda\) is nonzero. While not incorrect, we avoid the reader perhaps having to think about the technicality.
After Exercise 1.4.7, when mentioning what the group generators are, be more precise to make sure we don't include the "dilation by 0".
When invoking the real inverse function theorem before Theorem 2.2.8, give the derivative formula for all \(q \in V\) rather than just \(p.\) That's the way we state the holomorphic version too and that's what's in appendix B.
A little bit more detail on the second dot product in the proof of Proposition 2.2.9.
Be explicit in Exercise 3.1.1 that \(k\) is an integer (though it should be clear from context).
In Exercise 3.1.12, specify that \(n\) is nonzero to avoid having to work through the technicality of the zero chain (where the \(\gamma\) is not a "path" according to our definition.
In Exercise 3.2.25, in the hint about \(dA\) being the area measure it should steer the student towards polar coordinates, but \(r\) is already used, so write it as \(\rho\, d\rho\, d\theta.\)
In Exercise 3.2.21, assume that \(f\) is nonconstant and to not have to be wordy assume that \(U\) is a domain to fix an erratum.
In Exercise 3.3.36, assume that \(f\) is nonconstant to fix an erratum.
In Proof A of Lemma 3.4.1, note explicitly why \(h\) is continuous by referring to the proposition B.2.1 in the back.
In Exercise 3.5.9, be a little bit more pedantic and explicitly say that the points are distinct.
Reword Exercise 3.5.5 to make it clearer and in the proof above mark where it is used with a parenthetical.
In the proof of 4.3.4, mention very fleetingly that the constant is nonzero (it is obvious), but making it clear why we can add a constant to the exponential.
In Exercise 4.5.4, allow rescaling the interval in part b, and avoid saying it is a path, as it is not for \(n=0\)
In the proof of 5.2.2 (Riemann extension) explicitly mention that we're assuming we are not in the trivial case when \(f\) is constant. Same in the proof of (ii) in Corollary 5.2.3, and also in the paragraph below Definition 5.2.4.
When defining holomorphicity at infinity, do not depend on the convention just defined above as we said we will not usually use it, and just mention that we consider \(\frac{1}{f(1/z)}\) if it was a pole.
The path in Exercise 5.3.10 is corrected, the old one did not lead to a solution!
Improve the explanation of why argument principle is called the argument principle.
Exercise 5.5.1, the codomain of \(f\) is now \(V\) as it should be.
In Exercise 6.1.2, do not assume that the limit is bounded, that follows quickly and really should be proved by the student. Last time I forgot to remove this element of the exercise when it was changed to only consider a sequence of bounded functions.
In the proof of Montel, flip the definition of \(r\) and the bound. What was there is true, but requires more thinking, just get the \(r\) out of the definition directly.
In the proof of Theorem 7.2.3, say \(M > 0\) is an upper bound rather than the supremum to avoid having to deal with the \(M=0\) technicality.
In the paragraph before Theorem 7.3.2 (Bôcher), be a bit more explicit to better align with the theorem statement. Make the naming also align.
In the proof on page 185, mention that we can just assume \(A=0\) as we could have just picked any particular \(\Phi\) to begin with.
In Exercise 7.3.1, change the "bounded" hypothesis to "bounded from below" which is more natural (bounded was a typo) here and should point the student better in the right direction anyway. The exercise is "backward compatible" in the sense that a proof for "bounded" using Bôcher as asked for works just as well for this weaker hypothesis.
After the statement of Theorem 7.3.3 (Schwarz reflection), add a sentence about using biholomorphisms.
Shrink Figure 7.6 vertically a tiny bit.
In the proof of Schwarz reflection for harmonic functions (7.3.3), mention explicitly that \(F\) is clearly continuous on \(U.\)
Reword Exercise 7.3.6 so that it doesn't seem like we are asking for existence (that was proved before).
Add a hypothesis to Exercise 7.3.7 to fix an erratum. The function \(f\) must to be continuous.
Make the discussion of subharmonic functions a bit less wordy, more precise and hopefully more understandable.
In the proof of sub-mean-value property, correctly handle the case when the Darboux integral is \(-\infty\) and change the wording above to make it clear (also simplify the paragraph about step functions and continuous functions).
Slightly reduce wordiness of the proof of Radó.
In the definition of convergence of products, mention the inconsistency of definitions in the literature and the "diverges to 0" terminology.
Simplify the bit about sum of principal Logs not quite being the principal Log of the product in the proof of Proposition 8.1.3. Also mention explicitly that we can simply assume that \(a_n\neq 0 .\)
Added a note about what we mean by "converges absolutely to L" to avoid possible confusion.
Proposition 8.1.7 was missing some hypothesis, so added that the series is uniformly bounded, and left the way it is used (compact \(X\) and continuous \(g_n\) as a new remark after the proposition.
Split Exercise 8.1.1 into 3 parts that guide the student. The proposition we were proving had an erratum anyway, so the original exercise was not possible, and it would be a bit difficult without guidance.
Before the Weierstrass theorems in chapter 8, note that their finite versions are obvious as well, just for completeness.
At the end of the proof of Lemma 9.2.1, mention explicitly that we must divide by \(2\pi i.\)
In Exercise 9.2.1, add the missing hypotheses that the functions in the algebra are bounded.
In the proof of Corollary 9.3.6, add a short parenthetical remark about how the theorem shows that \(U \cup \widehat{K}\) is open.
At the end of the proof of Mittag-Leffler, split of the \((\ell+1)\)th term too and both simplify and make more precise the reason why everything but \(f_\ell\) is holomorphic at \(p .\)
In Exercise 9.4.2, use \(m\) for the degree of the geometric sum to avoid making the student think that it is the same \(n\) as the index in the sequence.
Replace Exercise 9.4.2 with a simplified version to fix an erratum. Assume that the \(U\) in Weierstrass product theorem is simply connected and give an outline of the proof.
Before Definition 10.1.2, make it more precise to say that the real-analytic function has nonvanishing derivative (that follows from the actual definition).
In Corollary 10.1.4, and therefore Exercise 10.1.1, the conclusion is only that \(F\) is meromorphic (obviously poles where \(f\) had zeros).
In the basic analysis results section, do a few minor changes and make the pagination be reasonable again after an update to LaTeX.
Add Cauchy's theorem for derivatives and for polynomials to the index, and in general clean up the index a little.
Some minor clarifications and many minor style and grammar fixes.
Fix the known errata.

November 5th 2025 edition (version 1.8):

The main focus of this version is to fix errata and other issues identified with the help of AI (no content was generated by AI).

In the proof of Theorem 3.2.9, we handle the case when \(g(z_0)\not=0,\) but that never happens by definition of \(g,\) so just remove that sentence.
In Exercise 3.2.20 it now correctly says \(d\bar{z} = dx - i \, dy .\)
In Exercise 3.4.10, the exponential order definition in the exercise now correctly says \(|f(t)| \leq M e^{ct} ,\) though that changes the exercise (to what was intended originally obviously).
In the proof of Proposition 3.5.3, use \(b\) instead of \(a\) in the proof, as it is not quite the \(a\) from the statement.
Reword Exercise 4.1.7 to be clearer.
Rewrite the proof of Lemma 4.7.2 to hopefully be clearer.
In the proof of Theorem 4.7.3, make the lines be horizontal to avoid a rotation midway through the argument. Modify Figure 4.10 to match this new setup. It is true that it makes the first part a little weird as it's not the graph \(y=f(x)\) but \(x=f(y)\) now, but I think it is now clearer and matches the picture in all parts of the proof.
Be more careful with the parenthetical remark about simple pole in Proposition 5.3.5, \(f\) only has a simple pole at \(p\) if \(h(p) \not=0 ,\) so just say "at most a simple pole at \(p\)".
In Exercise 6.1.2, the functions in the sequence are now assumed to be bounded, otherwise one would only get the result for a tail of the sequence.
Replace the faulty Exercise 6.2.8 with a different one.
In Exercise 6.3.2 part c, the definition of the upper half disc is now fixed.
In the proof of Theorem 7.3.2, make the argument about \(-c_1 \geq 0\) a tiny bit clearer.
In the proof of Theorem 7.4.3, emphasize that \(f= \varphi+h \) when using it in the long estimate.
Fix the erratum in the proof of Proposition 8.1.3 by replacing the proof by a simpler one. Also put the note about the sum of Logs into the proof as we need it there.
Add a short note before Corollary 8.1.9 about the change in notation from \(\prod (1+g_n(x))\) to \(\prod f_n(x)\) and what it means in terms of the definitions.
In section 9.4, try to use letters as indices only for specific purpose, to make things clearer. In particular, use \(m\) for the index in the expression \(P_p,\) in the statement of the theorem, and use \(p\) for the point in Example 9.4.2 and in the statement of Exercise 9.4.1.
Add Example 9.4.3 with the example from Exercise 9.4.2, showing that the \(R_n\) really are necessary in general.
Add a note after proof of the Monodromy theorem on page 229 about all paths fixed-endpoint homotopic in a simply connected domain, and add an exercise (10.2.17) to prove the converse (actually prove existence of primitives) as it is a nice quick application of continuation.
Fix the known errata.

May 20th 2025 edition (version 1.7):

A couple of very minor language improvements.
Fix the known errata.

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.

Make the caption in Figure 1.5 more correct: The sphere is the normal \(S^2\) sphere.
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.
Write Proposition 2.2.5 even for \(n=0\) but add the special case to the formula then. This 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.
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.
On page 36, add a couple of sentences about why we have the modulus squared in the determinant.
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 is good practice as well as being useful in another exercise before we get to the formula with the derivatives.
On page 54, don't tell the reader to necessarily use sines and cosines, and mark Exercise 3.1.1 as easy.
Add assumption that \(g'\) is continuous to Proposition 3.1.7 as we haven't yet proved it, and add a footnote about it.
On page 58, note that the caveat about convergence of paths applies to both the \(dz\) and the \(|dz|\) integrals.
Move Figure 3.3 up a paragraph, which makes for a much nicer page break and moves Cauchy-Goursat to next page.
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.
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.
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.
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 \(K' .\)
On page 87, add a comment about Schwarz-Pick providing the bound at other points than 0.
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.
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 simple connectedness.
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.
Make the hint in Exercise 4.4.1 be more useful; pointing the student at subsection 3.4.2 is a better hint.
Add footnote to Definition 4.5.9 to note that this is the actual definition of homotopy for any path connected topological space.
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.
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.
In Example 5.3.6, show the use of all three propositions for computing the residue.
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).
Add a remark after statement of Lemma 5.6.2 that the image of the disc is a neighborhood of \(f(p).\)
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.
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.
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.
On page 180, the last displayed inequality in the proof of Harnack is actually an equality.
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.
In Figure 7.6 stop marking \(U_-,\) which we never defined.
Clarify the proof of Rado's theorem.
Fix the known errata.

July 19th 2023 edition (version 1.5):

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.
Consistently use "converges uniformly on compact subsets" instead of sometimes "converges uniformly on compact sets."
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.
Add Exercise 6.3.9 (entire and injective implies onto), a nice application of RMT.
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 \(n .\)
Change the proof of 7.1.10 (Identity) to be more consistent with 7.1.11 (Maximum principle).
Fix the known errata.

May 16th 2023 edition (version 1.4):

Change "up to multiplicity" to "counting multiplicity." We were using both so stick to one, and the second one is clearer.
In the description of the typical application of Hurwitz, be a bit more precise (besides fixing an erratum).
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.
A few other minor language/style improvements or clarifications.
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.

In the proof of Proposition 3.1.8, mark which integral is the arclength integral of the modulus.
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.
In the intro to 3.3.2, the example of real differentiable function would only work for bounded \(g\) so integrate from some \(c \in (a,b).\)
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.
Streamline the wording of Exercise 3.3.19 a bit and include the definition of \(\limsup_{z \to \infty} .\)
Change the wording of Theorem 3.3.10 (Liouville) to be more precise.
Improve the wording of proof of Theorem 3.3.11 (FTA) a bit more.
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.
Note that Exercise 3.5.3 is called the Cartan's uniqueness theorem and add it to the index.
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.
In proof of Corollary 5.2.3, emphasize where proof of first item ends and the proof of the converse statement starts.
In the comments after Definition 5.2.4, emphasize that \(\ell \in \mathbb{Z} .\)
Reword Exercise 5.2.23 to be a bit more readable.
Reword Exercise 5.4.6 to make it explicit as to what the power sums are.
Reword proof of Exercise 5.4.7, it is a bit misleading.
Simplify statement of Hurwitz a tiny bit.
In Exercise 6.2.2, emphasize that \(U \subset {\mathbb{C}},\) as the two notions are not the same in an arbitrary metric space.
Reword part a) of 6.2.6 to be more explicit.
In Theorem 7.4.5 (maximum principle for subharmonic functions), add a footnote to draw attention to the maximum now being a global one.
Reword parts b and c of Exercise 9.2.7 to be more logical.
Add some more hyperlinks.
Many other minor clarifications and cleanups.
Fixed many misspellings, grammar and style issues.
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.

Add polar form and polar coordinates to the index.
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.
Also on page 18, use \(w\) for the target variable to avoid confusion when defining the annulus and the sector.
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.
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.
In subsection 2.1.1 emphasize a tiny bit more the fact that the \(h\) is complex.
In subsection 2.1.1 explicitly state what \(o(|h|)\) means rather than being vague.
Reword the paragraph in front of Proposition 2.1.4 to be clearer.
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.
At end of subsection 2.1.2 emphasize the exercise for holomorphicity of the exponential.
In the first proof of Proposition 2.2.2, handle the \(k=0\) case explicitly.
In the 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.
Add some segue sentences in subsection 2.2.1, and rename Proposition 2.2.5 to "Power rule and its consequences."
Exercise 2.2.2 should just ask for the power rule, the second item is asked for in Exercise 2.2.3.
In Exercise 2.2.11, just use the identically equal sign to make it easier to read.
Add a note about what Proposition 2.2.9 means in view of Exercise 1.1.7 after the proof of the proposition.
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.
Proper proof of the Cauchy-Hadamard theorem, it seemed to imply divergence about the series of the absolute values.
Add the word "absolutely" to Figure 2.4.
Reword Remark 2.3.5. We do definitely need addition, it is multiplication that could plausibly be put off for later.
In the proof of 2.4.6, save the "the series converges" only after we proved what it is.
Emphasize the idea of factoring out the zero out of a power series after the proof of the identity theorem (2.4.7).
In Exercise 2.4.17 emphasize to only show uniqueness.
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.
After the definition of the line integral, note that the definition still holds even if the derivative would be zero.
In simple justification of reparametrization, use \(h' > 0\) and \(h' < 0\) instead of increasing decreasing to avoid zero derivative.
In Exercise 3.1.5, say \(f\equiv 1\) instead of \(f=1\) for emphasis.
Fix Proposition 3.1.7 to mention that the new path might not be a path if we are disallowing zero derivatives.
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.
In proof of 3.2.9 make sure to state that \(\alpha\) is a complex number.
In proof of Theorem 3.3.11, no need to handle the case when \(P(z)\) is constant, it is nonconstant by definition.
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 \(M.\)
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).
Reword Definition 3.4.4 to read more naturally hopefully.
In proof of 3.4.5, one of the equalities was written as an inequality so fix that (it was not really wrong).
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.
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 \(\log (z-p).\)
In Definition 4.2.1, use "open" rather than "domain", connected is not needed.
Improve the wording of proof of Theorem 4.2.2, it was misleading in a couple of bits.
Emphasize before Proposition 4.3.7 that simple connectedness is a topological concept.
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.
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.
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-\(C^1 .\)
Replace the quote at the beginning of chapter 5 with one that irritates me less given what's happening in the world.
Tighten up section 5.1 a bit.
Add Exercise 5.1.7 to prove L'Hopital's rule.
In the paragraph after definition 5.2.1, give parenthetical example of a pole and an essential singularity.
In the proof of Theorem 5.2.2, say \(h\) is holomorphic on \(U\) rather than just "near \(p\)."
Exercise 5.2.8 was worded a bit vaguely, the function is supposed to be "not identically zero".
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.
Exercise 5.2.12, the \(f\) should be "not identically zero."
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).
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.
In Rouché, the winding number should be 0 or 1 for all \(z \notin \Gamma,\) rather than \(z \in U .\)
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.
Shortening/tightening some wording to improve the flow improves also the pagination a tiny bit in sections 6.2 and 6.3
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 \(f(z)-f(p).\)
Add Exercise 6.2.10, which is a nice application of Montel to characterizing the radius of convergence.
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.
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 \(g'(p).\)
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 \(n.\)
Add Exercise 7.3.11 that requires infinitely many reflections.
On the bottom of the first page of 9.1, we expand on why the series for \(f\) cannot converge on the square.
In the proof of the reflection principle (10.1.1), mention that \(F\) is continuous. That is rather immediate, but it should be stated.
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.
Before Corollary 10.1.3, remove the sentence about "switching sides" it is more confusing than helpful.
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 \(f(\partial\mathbb{D}) \subset \partial \mathbb{D}.\)
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.
Be more precise in the statement of Proposition 10.2.11.
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.
Flip the order of proof in Proposition A.2.13 to be more logical.
In Exercise A.3.11, only ask for the part of the proposition that was not yet proved to avoid confusing matters.
Theorem A.5.7 should mention that \(X\) ought not be empty.
A little bit of cleanup of the proof of Theorem B.3.16.
Clean up the proof of Proposition B.2.1 to not use sequences as it's simpler without.
Some minor clarifications and fixes throughout.
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.