Add proof of Cauchy-Goursat theorem, and various small changes

[user202729]

I'm obviously just using the book as my lecture note now…

Anyway, I think that saying "works very hard to prove it for the case of a triangle" is a really unfair description of the proof (the key idea is the "binary" (actually quaternary) search. Triangle, rectangle, or circle does not matter --- although it certainly is more difficult to do it in the case of a circle.) (Thinking about it, if you do it on a A4-paper-shaped loop γ, then you can do a binary search.)

And it isn't that difficult.

Plus some minor fixes. (There should be a backslash-space after \dots for consistency, but TeX strips trailing spaces anyway, so LaTeX defines backslash-newline to be the same thing as backslash-space.)

[vEnhance]

I'm obviously just using the book as my lecture note now…

Great! :)

Anyway, I think that saying "works very hard to prove it for the case of a triangle" is a really unfair description of the proof (the key idea is the "binary" (actually quaternary) search. Triangle, rectangle, or circle does not matter --- although it certainly is more difficult to do it in the case of a circle.) (Thinking about it, if you do it on a A4-paper-shaped loop γ, then you can do a binary search.)

And it isn't that difficult.

I think a long time ago I had the impression the proof was difficult. Seeing it written like this confirms I was wrong. So yeah, I think this is an obvious improvement.

[user202729]

Speaking of which:

• That chapter has an inconsistency of notation: sometimes a domain (simply connected open set) is denoted by U, sometimes it's denoted by Ω. (Or so it seems, I haven't carefully checked if U or Ω is required to be simply connected yet) Which do you prefer?
• Normally section title is named "Bonus", but that section title I named it "Optional"… it's inconsistent.

[user202729]

Actually this change may backfire... after all (n) is defined to be {n \alpha | \alpha \in \OO_K}. Guess I'll just revert it.

[user202729]

Which feels slightly silly, but I don't see much harm either.
Without defining Dedekind domain, the theorems would read

• 𝒪_K is Noetherian, integrally closed, and "dim Spec 𝒪_K = 1" i.e. prime ideals are maximal.
• ⟹ prime ideal factorization works in 𝒪_K.

[vEnhance]

Seems fine.

[vEnhance]

Speaking of which:

• That chapter has an inconsistency of notation: sometimes a domain (simply connected open set) is denoted by U, sometimes it's denoted by Ω. (Or so it seems, I haven't carefully checked if U or Ω is required to be simply connected yet) Which do you prefer?

I think it's supposed to be Omega for simply connected and U if not necessarily simply connected. Any violations of that guideline should be fixed.

• Normally section title is named "Bonus", but that section title I named it "Optional"… it's inconsistent.

I think it's fine to label this section as "Optional". The convention is right now

• Bonus for entire chapters
• Optional for sections or smaller things.