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Tasty Bits of Several Complex Variables: Changes

June 3rd 2024 edition (version 4.1):

This is a minor new version incorporating some errata fixes, and several minor changes due to some feedback from Richard Lärkäng who taught with the book in the past semester, and a couple of other minor changes (some that I was planning to make for version 4.0 but somehow they slipped through the cracks).

  1. Instead of the functional notation for plugging vectors into differential forms, use the angle bracket pairing notation that is more standard and since we don't really use this much at all, there is no reason to have a less standard notation. This affects one place in chapter 1, exercise 4.4.3, and one place in appendix C.
  2. The notation \(\C \otimes T_p M\) is changed to the more common \(\C T_p M\) which also avoids a slight abuse of notation. Thanks to Richard Lärkäng for the suggestion.
  3. Lemma 2.3.9, the statement about the Levi form doesn't really make sense if \(M\) is not a boundary, so change the statement to explicitly say that the inside is "above" the \(M\) in the new coordinates. Also explicitly mention this at the end of the proof. Thanks to Richard Lärkäng for the suggestion.
  4. At the beginning of section 2.4, give a reference to the one-variable book as not every student may have had a treatment of harmonic and subharmonic functions in a one-variable course.
  5. Add hints to Exercise 2.4.8.
  6. Change Proposition 2.4.5, to be about the sub-mean-value property for all small enough discs (which is really the best way to prove the original statement in the first place), leaving the original statement as an "In particular". That also changes Exercise 2.4.11 by really making it a bit easier as it gives a way to do it (and change the hint to that exercise too). Thanks to Richard Lärkäng for the suggestion.
  7. In Example 2.5.4, put a square on the norm of \(z\) as that makes the computation easier. Thanks to Richard Lärkäng for the suggestion.
  8. In Theorem 2.5.6, add part (iv) which is just the conclusion of the second version of the Kontinuitatssatz as that just follows from the proof directly with no extra work. Thanks to Richard Lärkäng for the suggestion.
  9. In Theorem 2.5.6, make part (ii) less wordy: "U is Hartogs pseudoconvex". I mean the definition is just above the theorem!
  10. Mark Exercise 2.6.1 as Oka's lemma to make the connection to the "proof" above clear.
  11. Change Exercise 4.4.1 to ask also to prove the Leibniz rule (which easily follows from the first part and the real Leibniz rule), but this is good to understand for the computations later in the section. Thanks to Richard Lärkäng for the suggestion.
  12. Add Leibniz rule to Appendix C.
  13. Simplify the wording of some of the theorems in Appendix E.
  14. Fix errata.

December 9th 2023 edition (version 4.0):

This is a major new edition with some new material and many smaller changes throughout. Important: Exercise numberes may have changed!.

Larger changes in general:

Detailed list of the smaller changes.

December 23rd 2020 edition (version 3.4):

May 28th 2020 edition (version 3.3):

October 1st 2019 edition (version 3.2):

May 21st 2019 edition (version 3.1):

May 6th 2019 edition (version 3.0):

The main goals of this revision are:

The primary changes are listed below. These are only the somewhat larger changes. Lots of small minor improvements have been made throughout.

October 11th 2018 edition (version 2.4):

  1. Change definition of holomorphic to be the more standard one ($f$ locally bounded and complex differentiable in each variable, giving the Cauchy-Riemann definition only for continuously differentiable functions as an alternative) this doesn't require a deeper one-variable result (that would not appear in a basic one-variable book). Exercise 1.1.4 now makes more sense with this definition.
  2. Be a bit more formal with the definition of one-variable holomorphic function, and define it with the complex derivative to make the one variable section aligned with the new definition in the first chapter.
  3. Be more explicit and careful with "uniformly absolute" convergence of series and when meaning "uniformly absolute convergence" always state it that way (even though we say that the only type of convergence in several variables is absolute, so it can't mean anything else).
  4. Add definition of $O(\ell)$ when first used.
  5. All links are https now.
  6. Minor clarifications and style fixes throughout.
  7. Fix errata.

June 27th 2018 edition (version 2.3):

  1. Fix Exercise 1.2.11, and add part c).
  2. Fix Exercise 1.4.1, and add part c).
  3. As people have different opinions about what "strongly (pseudo)convex" means for unbounded domains, only define the term for bounded.
  4. Improve exposition very slightly throughout.
  5. Fix errata.

November 29th 2017 edition (version 2.2):

  1. Reword the hypotheses of proposition 6.2.5, 6.2.6 and theorem 6.2.7
  2. Move exercise 4.3.4 to section 4.2 so renumbered to 4.2.4. It makes a lot more sense right after 4.2.3. (Thanks to John Treuer for the suggestion)
  3. Improvements and fixes in language and style in a number of places.
  4. Fix errata.

March 21st 2017 edition (version 2.1):

  1. Add exercise 6.4.21
  2. Fix wording of exercise 6.4.16, $\ell_j$ are not needed
  3. Fix definition of meromorphic functions to be the standard one, noting the deep result of Oka that shows what localy a quotient means globally a quotient on domains in ${\mathbb C}^n$.
  4. Very minor improvements in style and exposition in a few places
  5. License changed to dual CC-BY-SA and CC-BY-NC-SA
  6. Fix errata.
  7. Fix few minor typos

May 5th 2016 edition (version 2.0):

November 24th 2015 edition:

August 21st 2015 edition:

November 19th 2014 edition:

Improvements in exposition in a bunch of places and also fix errata.

September 2nd 2014 edition:

May 1st 2014 edition:

First version

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