# Tasty Bits of Several Complex Variables: Errata

Every book (no matter how much you paid for it) has errors and typos, especially text that is new in a given edition. I try to be as transparent as possible about any errors found, and I try to fix them as quickly as possible. Please do let me know if you find any errors or typos, so that they may be fixed. A free book relies on readers sending in corrections.

This page lists major problems in the various editions. Simple typos, misspellings, and such are not listed here. Also not listed are things that were correct but simply deserved improvement. Of course, older editions may suffer from recently discovered errata as well.

October 1st 2019 edition (version 3.2):

• Page 42, proof of Corollary 2.1.5. The point $p$ should be $(0,\ldots,0,\frac{3}{4})$, and $m=n-1$ and $k=1$. Thanks to Sivaguru Ravisankar.
• On top of page 56, Example 2.3.7, in definition of $z_j$ it should be $Z_j$, not $Z_1$. Thanks to Nicholas Lawson McLean.
• On page 65, Theorem 2.4.2 is stated for complex-valued functions. While that's not wrong, it is inconsistent, in the section we only consider real-valued harmonic functions.
• On page 157, in the explanation of the limit going to infinity at the top of the page it says "as $|z| \to \infty$" but it should say "as $|z-p| \to 0$."

May 21st 2019 edition (version 3.1):

• Page 18, the formula for $c_{\alpha}$ has a $z$ instead of $a$ in the denominator. Thanks to Alan Sola.
• Page 121, on the bottom of the page the $U$'s should be ${\mathbb D}$ since we're computing the Bergman kernel for the unit disc, not just any $U$.
• Page 122, Exercise 5.2.6: This exercise appears to be way too hard. It is enough for just the unit ball or polydisc. And in fact, it doesn't apply for the unit polydisc where it may be useful in a later exercise. So it should be replaced with $U$ being either the unit ball or the unit polydisc.
• Page 129, when writing $\sigma_2$ as a polynomial in the power sums at the bottom of the page, there are missing parentheses, the $\frac{1}{2}$ should multiply the entire expression, not just the first term. Thanks to Achinta Nandi.
• Page 155, Appendix B, Lemma B.12 (Schwarz's lemma): The function ought to be valued in the unit disc, that is, $f \colon {\mathbb D} \to {\mathbb D}$. Thanks to Trevor Fancher for pointing this out.

May 6th 2019 edition (version 3.0):

• In Exercise 1.1.7, assumption that the triangle lies in a complex line was missing. Thanks to Trevor Fancher for pointing this out.
• On page 19, Proposition 1.2.2, where it says "Cauchy estimates" it should be an inequality, not an equality. Thanks to Trevor Fancher.
• In Exercise 1.4.2 part c, the $\Delta \cap {\mathbb B}_n$ should be $\Delta \cap {\mathbb B}_2$. Thanks to Nicholas Lawson McLean.
• On page 32, in the definition of "... the degree $d$ homogeneous part ..." it ought to say "... the degree $j$ homogeneous part ..." Thanks to Nicholas Lawson McLean.
• In Exercise 3.2.3, it should say "Prove that $X_p f$ is well defined". The $f$ was missing.
• On page 94 in the proof of proposition 3.2.8, in the first paragraph the arguments of $r$ are ordered inconsistently with the rest of the proof, it should be $r(z,\bar{z},w,\bar{w})$.
• On page 155, Schwarz's Lemma, the assumption $|f(z)|=|z|$ must be true for some nonzero $z$ for $f$ to be just a rotation.

October 11th 2018 edition (version 2.4):

• On page 6, when writing down the Cauchy-Riemann equations $u$ and $v$ are the real and imaginary parts of $f$.
• On page 7, in the computation of $\frac{\partial f}{\partial z}$ when computing as the $y$ derivative, the $i$ should be really $\frac{1}{i}$ or $-i$. That is $\frac{1}{i} \left( \frac{\partial u}{\partial y} + i \frac{\partial v}{\partial y} \right)$.
• In Theorem 0.1.1, the Cauchy integral formula, we forgot to require that $U$ is bounded (that is, it's not the "outside" of a Jordan curve). The setup is correct in Theorem 4.1.1 (Cauchy-Pompeiu).
• In Theorem 0.1.2, the absolute value signs are missing on the right hand side. While what's there is true and equivalent to regular maximum principle, it would be an odd statement of it.
• In section 1.2, to interpret the computations correctly, you need to interpret $\frac{z}{w}$, when $z$ and $w$ are in ${\mathbb C}^n$, as $\frac{z}{w} = \left( \frac{z_1}{w_1}, \ldots, \frac{z_1}{w_1} \right)$, and this is not listed in the list of notations at the beginning of the section, and it should be.
• On page 16, it says we are summing over $\alpha \in {\mathbb N}_0$ and that there is no natural ordering on ${\mathbb N}_0$. That should be ${\mathbb N}_0^n$ in both instances, as it says in the actual formula.
• On page 17, best not to try to interpret the second line in the computation, the third line follows directly from the computation above, so it's not needed, and it would require extra notation.
• On page 20, in the identity theorem (1.2.5), it should say "a nonempty open subset $N$" rather than "an open subset $N$".
• On page 25, in the holomorphic implicit function theorem, it says $f$ is a holomorphic mapping to ${\mathbb R}^m$ which is nonsense, it should say ${\mathbb C}^m$. Thanks to Trevor Fancher.
• On page 26, in the Riemann mapping theorem, the domain should be nonempty.
• In the proof of "Riemann extension theorem" (1.6.1), it says $|z_n|^2=r$ when that should be $|z_n|=r$.
• In the proof of theorem 1.6.2, it is not really clear that $h$ is a derivative of order $k-1$ so that it vanishes identically on $N$, but has a first order derivative that does not.
• In Exercise 2.1.2, take the interior of the intersection, so that you have an open set.
• In the example on page 52 it says ${\mathbb B}_n$ is defined by $\|Z\|=1$. It should say $\partial {\mathbb B}_n$.
• On page 56 in the proof of Lemma 2.3.8, after the second equals sign in the long computation, the third term (the one with denominator 4) is missing a factor of $a$. The end result of that computation is correct.
• On page 58, proof of Theorem 2.3.10 (tomato can principle), it says "strict minimum" below the second displayed equation. It should be "strict maximum".
• On page 67, just below the displayed equation in proof of Radó, it says "Fixing t" when that should be "Fixing z".
• Yaikes, the proof of Proposition 2.4.2 (which is an exercise) requires the solution to the Dirichlet problem in the disc which is Theorem 2.4.9. We will have to reshuffle this in the next version.
• In proof of Proposition 2.4.3, the following sentence is missing in the beginning of the proof: "Suppose $f$ attains a maximum at $a \in U$."
• In the proof of Theorem 2.4.8, when $f$ is not bounded below we replace it with $\max \{ f , -1/\epsilon \}$. The minus sign is missing.
• In Theorem 2.5.2, the continuity principle, it is not explicitly stated that the closed discs are subsets of $U$, although it is clear from the proof.
• In the proof of Thereom 2.5.6, we say "Let $u = \operatorname{Re} f$ for a holomorphic function $f$." Which is not possible for every $u$. One needs to first state that it is enough to show the inequality for $u$ which are say harmonic on a neighborhood of the closure $\overline{\Delta}$.
• On page 70, proof of Theorem 2.5.6, at the bottom of the page it should say $\varphi_t(\Delta)$ not $\varphi_t({\mathbb D})$.
• On top of page 74, when it says "let $M \geq 0$ be the operator norm of the Hessian matrix of $r$", it should say "let $M$ be an upper bound on the operator norm of the complex Hessian of $r$ near $q$".
• On page 84, in Proposition 3.2.2, it says "near some point we write", that should say "near any point if we write".
• On page 86, in the proof of Proposition 3.2.8 the last statement is not so simple (there is a gap in the proof). The vector field indeed annihilates $\Phi(z,\bar{z},w)-\bar{w}$, but to be tangent it needs to annihilate a real defining function, such as the real part of that, so it also needs to annihilate $\bar{\Phi}(\bar{z},z,\bar{w})-w$.
• In a related typo in Proposition 3.2.8, $\Phi$ is not $O(2)$ really, it is weighted-O(2) with $w$ having weight two. In particular, it is zero at zero and all the $z$ and $\zeta$ derivatives are zero.
• In Proposition 3.2.11, $f=0$ on a nonempty open subset of $\partial U$.
• In exercise 3.4.7, this should be all one sentence, and the extension (that does not exist) is a "holomorphic" extension of course. (Thanks to John Treuer)
• On page 104, exercise 4.3.2 the boundary must be connected like in Severi's version of this theorem.
• On page 104, exercise 4.3.3 is not stated quite right and does not quite make sense as stated. Best to just skip it.
• On page 116 in Exercise 6.1.3, the wording is not quite right. If the wording becomes more precise the problem becomes rather trivial. The point is that $g$ can be analytically continued from $p$ to $U$, not that the domain of $g$ can be made larger to include $U$, that is not true.
• On page 116 in Exercise 6.1.4, in part b, the ideal is an ideal in the ring of germs at the origin, and in part c, $p=0$. Or otherwise the $m$ would have to be defined by $(z_1-p_1,\ldots,z_n-p_n)$.
• On page 119, in the definition of the function $u$ using the Cauchy formula the $\frac{1}{2\pi i}$ is missing.
• On page 126, in proof of Theorem 6.4.5, it says assume $U'$ and $D$ are small enough so that $X$ has no limit points on $U' \times D$, that should be $U' \times \partial D$.
• In exercise 6.4.4, the inclusion at the end is wrong, you need to prove $I_p\bigl(V(I)\bigr) \supset I$. (Thanks to John Treuer)
• In beginning of the proof of 6.4.5, "has to a germ" should be "has to exist a germ". (Thanks to John Treuer)

June 27th 2018 edition (version 2.3):

• The definition of holomorphic functions (Definition 1.1.2) is correct (so it's not really an erratum), but it is not quite compatible with most introductory one variable books. This was not on purpose. I will change the definition to be the more standard one using the complex derivative in each variable, and when using Cauchy-Riemann we will assume $C^1$. Very little is lost by the reader just assuming that $f$ is continuously differentiable in the definition.
• In the footnote on page 82, the polarization identity is only for the real one (only the real part). It should say $4 \langle z,w \rangle = {\|z+w\|}^2-{\|z-w\|}^2 +i \bigl( {\|z+iw\|}^2 - {\|z-iw\|}^2 \bigr)$. What's there is true for real $z$ and $w$.

November 29th 2017 edition (version 2.2):

• Exercise 1.2.11 part b) is only true if $n=2$. In the next update there will be a new part c) which asks to find a counterexample if $n=3$. (Thanks to Liz Vivas for pointing this out)
• Exercise 1.4.1 part b) is wrong: There can be a one-real-dimensional curve of such points, but there can be no neighborhood of $p$. A better way to state the problem is to prove that if $\Delta$ is an anlytic disc, prove that $\partial \mathbb{B}_n \cap \Delta$ is nowhere dense in $\Delta$. (Thanks to Alekzander Malcom for pointing this out a couple of years ago, and I forgot to mark it down, and now thanks to Trevor Fancher for finding this issue again.)
• Exercise 6.4.21: an analytic disc in ${\mathbb C}^2$ is only a local variety (the word "local" was missing), since it is not necessarly closed in ${\mathbb C}^2$.
• Exercise 6.5.1: replace "hypersurface" with "submanifold".
• In the discussion of Segre variety on page 133, where it says "(and nowhere else)", that is meant only if $\widetilde{r}$ is a defining function in the same sense as $r$.
• In Example, 6.5.3, when setting $\bar{z}=\bar{w}=0$ we forgot to actually set $\bar{z}=0$.

March 21st 2017 edition (version 2.1):

Thanks to John Treuer for pointing out the following errata:

• On page 45, in the $2 \times 2$ Hessian at the top, the lower right derivative should be $\frac{\partial^2 r}{\partial y^2}$.
• On page 77, about midway down, the series is written $\sum_{\alpha} \frac{1}{\alpha} \cdots$ when it should be $\sum_{\alpha} \frac{1}{\alpha !} \cdots$. This is just the Taylor series for $f$.
• On page 80, $f_d$ is the degree $d$ homogeneous part (or $f_j$ is the degree $j$ homogeneous part).
• On page 85, in the proof of Proposition 3.2.6, "If both $M$ and $f$ are real-analytic", the $f$ should be a $\phi$, since we're trying to prove that $f$ is real analytic.
• On page 87, last line in proof of 3.2.10, "If $X_{q_k} f = 0$ for" should be "Then $X_{q_k} f = 0$ for"
• On page 100, exercise 4.1.3 is wrong (a bonus exercise is to find a counterexample to it as is currently stated using the unit disc as a domain), the correct hypothesis is not that $\int_{U} \frac{\frac{\partial f}{\partial \bar{z}}(\zeta)}{\zeta-z} \, dA(\zeta) = 0$ for every $z \in \partial U$. Also take $U$ to be bounded.
• On page 100, in exercise 4.1.4, the $\varphi$ in the integrals should be $f$.
• On page 116, in exercise 6.1.4 a), $I$ should be $(f)$.
• On page 118, right after the statement of the Weierstrass Preparation Theorem, "degree $m$ monic polynomial" should be "degree $k$ monic polynomial".
• On page 120 and 121, Propositions 6.2.5, 6.2.6, and Theorem 6.2.7. $\mathbb D$ and $D$ are both used. The theorems work for any disc $D$, not just the unit disc.
• On page 121, in the first line of the proof of Theorem 6.2.7, "Let $U_m$ be the subset of $U$" should be "Let $U_m$ be the subset of $U'$".
Other errata:
• On page 34, in exercise 1.6.2, the holomorphic function had better be defined in a domain (a connected set).
• On page 115, formula (5.2), on the right hand side it should be $\sum_j \varphi_j(z)\overline{\varphi_j(\zeta)}$.
• On page 117, in the definition of Weierstrass polynomial, the sum (the $\Sigma$) goes to $k-1$ (it was missing the upper limit).
• On page 130, an overzealous spellchecker corrected hypersurfaces to hypocrites.
• On page 135, in the Diederich-Fornaess theorem, there is no need to talk about the dimension of $X$ in the statement of the theorem.

May 5th 2016 edition (version 2.0):

• On page 22, the definition of meromorphic functions is nonstandard. The standard definition is that $F$ is the quotient of holomorphic functions locally. That the definitions are equivalent for domains in ${\mathbb{C}}^n$ follows via a deep result of Oka, and our definition would be misleading when generalizing to complex manifolds. So the next version will use the standard defition. Thanks to Debraj Chakrabarti for pointing this out.
• In chapter 5, page 108 at the end of proof of the Bochner-Martinelli formula we wrongly assume that $f$ is holomorphic, and while that's an important case the theorem is stated for smooth functions. In the last estimate therefore we also get terms of the form $\frac{\partial f}{\partial \zeta_j}(\zeta)(\bar{\zeta}_j-\bar{z}_j)$ which can also be bounded by $Mr$, so the estimate still follows. Thanks to Anirban Dawn for pointing it out.
• Exercise 6.4.16, the numbers $\ell_j$ were not needed, assume they are all 1.

November 24th 2015 edition:

• On page 8, when it says why we can swap the order of summation, it says it is because of uniform convergence, when it should say uniformly absolutely convergent. Similar error is due to cut and paste on page 16.
• In exercise 1.2.4, the $f$ should be holomorphic in a neighborhood
• In exercise 1.4.2, X and Y should just be assumed locally compact Hausdorff, and for the extension $f(\infty) = \infty$.
• In proof of theorem 1.6.3, 1) the $\varphi$ need only be defined for $\xi = 0$. 2) The last equality shouldn't include $\det Dh$ since that does not make sense, it is $\det D\varphi$. (also $\varphi$ and $\phi$ were both used by mistake here)
• In Definition 2.1.1, the set $V$ should be assumed nonempty.
• In Definition 2.2.3, replace $T_pM$ with $T_p\partial U$.
• On top of page 44, there is a $\frac{1}{2}$ missing in front of the Hessian in the equation that should be $y = \frac{1}{2} x^t H x + E(x)$. Similarly a few lines down.
• Page 49, the the computation about 2/3 of the way down, when we plugged in for $w$ in $-\operatorname{Im} w$, the minus sign disappeared.
• Page 54, in the proof of Theorem 2.3.10 (tomato can principle). In the first displayed equation in the proof, the second $z_1$ is missing a conjugate sign. In the second displayed equation, the $\lambda$ mysteriously disappeared from the right hand side.
• Page 70, Exercise 2.5.7, clearly the points $p$ should be in $\partial U$.
• In Exercise 4.3.2, $f$ should be defined on $\partial U$ not $U$.

August 21st 2015 edition:

• On page 13, the boundary of the bidisk is $\partial {\mathbb{D}}^2 = (\partial {\mathbb{D}} \times \overline{{\mathbb{D}}}) \cup (\overline{{\mathbb{D}}} \times \partial {\mathbb{D}})$. The closures were missing.
• Also on page 13, Exercise 1.1.3. Part b) is not correct as stated. I will just remove part b) from next version.
• On page 24, in Rothstein's theorem and also in Exercise 1.4.4, the hypothesis of "holomorphic" is missing.
• On page 35, Exercise 2.1.2, it should say every connected component is a domain of holomorphy as the intersection could be disconnected.
Thanks to Jianou Zhang for pointing out most of these.

November 19th 2014 edition:

• Exercise 1.2.9: The sequence of functions must be nowhere zero as in the standard corollary to Hurwitz from one variable, otherwise there are easy counterexamples.
• Exercise 1.5.1: There is an extra not, that is, "if U is not unbounded" should be "if U is unbounded". The hint hopefully makes it clear what was meant.
• Page 45, definition of $T_p^{(1,0)}M$, the right hand side should have intersection with $T_p^{(1,0)} {\mathbb C}^n$ of course, not M. Same for (0,1) vectors.
• Page 31, at the end of proof of Theorem 1.6.1, Exercise 1.6.3 is used and was forgotten to be mentioned, that is the complement of $g^{-1}(0)$ is connected.

September 2nd 2014 edition:

These are mostly minor typos. There were also a bunch of english typos or hard to understand sentences that are now fixed.

• In Example 2.3.6, for $z_j$ (little $z$) the $j$ runs from $1$ to $n-1$, not $n$.
• Theorem 2.4.14 as stated only makes sense for proper subsets of ${\mathbb C}^n$, that is the boundary of $U$ better be nonempty.
• In proof of Theorem 2.4.16 The $j$ should run from 2 to $n-1$.
• In proof of Proposition 3.1.5, clearly $f$ cannot be defined where $z\zeta = -1$, not 1.