# Tasty Bits of Several Complex Variables: Changes

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):

• Add chapters 5 (integral kernels) and 6 (varieties).
• Split the harmonic (plurisubharmonic) section (2.4) from Hartogs pseudoconvexity (new 2.5). That of course moves 2.5 to 2.6.
• Add Rado's theorem to the section on harmonic functions section 2.5.
• Add a few corollaries to Hartogs phenomenon.
• Make implicit function theorem a theorem rather than just an exercise.
• Add exercises throughout, and reorder some exercises to fix dependencies between them.
• Do some minor reordering in the early chapters.
• Add more detail about orientation in ${\mathbb C}^n$.
• Add diagram for proof of the Riemann extension theorem.
• Add diagram for proof of the tomato can principle.
• Add diagram for proof of the local nature of Hartogs pseudoconvexity (Lemma 2.5.7)
• Add diagram for proof of Lewy extension theorem.
• Add more historical context in several places.
• Lots of small minor improvements in style and exposition, some minor reordering and reworking of proofs and a few extra clarification in many places.
• Add more terms to the index.
• Fix errata.

November 24th 2015 edition:

• Add exercises 1.6.5, 1.6.6, 1.6.7 about Jacobian conjecture.
• Fix errata and several typos.

August 21st 2015 edition:

• Improvements in exposition, a few better pagebreaks.
• Replace "vanishes to order k" with "is O(k)" as the former seems to be not uniformly agreed upon if it is O(k) or O(k+1), wheras the latter is unambiguous.
• Add definition of "automorphism" and "automorphism group"
• Fix errata and several typos.

November 19th 2014 edition:

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

September 2nd 2014 edition:

• Minor improvements in exposition.
• Add exercises 1.2.9 and 2.4.25 (which moves 2.4.25 to number 2.4.26).

May 1st 2014 edition:

First version