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Results tagged with cv.complex-variables
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user 290
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
23
votes
0
answers
651
views
Which proofs of the fundamental theorem of algebra are "essentially the same" vs. "essential...
The classic MO thread Ways to prove the fundamental theorem of algebra contains $60$ proofs of FTA, and I'm sure there are many more in the literature. It would be nice to have some way to organize th …
- 118k
30
votes
1
answer
2k
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Which of the proofs of the fundamental theorem of algebra can actually produce bounds on whe...
One of the old classic MO questions is a big-list of proofs of the fundamental theorem of algebra. Here is a second big-list question about this big list:
Which of the FTA proofs can, even in prin …
- 118k
8
votes
1
answer
588
views
Defining holomorphic functions in terms of Banach algebras, and similarly for C*-algebras
Let $C$ be the category of commutative Banach algebras and let $U : C \to \text{Set}$ be the usual forgetful functor. The holomorphic functional calculus guarantees that every holomorphic function $f …
- 118k
12
votes
1
answer
2k
views
Wick rotation and the Riemann zeta function
The goal of this question is to conceptualize in some way the fact that the Riemann zeta function $\zeta(s)$, and other zeta functions like it, have analytic continuations.
Background
I have by now …
- 118k
17
votes
3
answers
2k
views
What is a reasonable finitary analogue of the statement that harmonic functions are smooth?
In my answer to this question on MU, I suggested that the OP think about the difference between real-differentiable and complex-differentiable functions by using a sort of finitary analogue. One way …
- 118k
13
votes
1
answer
859
views
What does the incidence algebra of the lattices in C tell us about modular forms?
I have two different and probably unrelated questions that can both be superficially described by the title, so I hope you'll forgive me if I ask them together. They both fall under the category of t …
- 118k
7
votes
1
answer
446
views
Reference for equivalent definitions of the genus
Let $X$ be a (edit: nonsingular) projective complex algebraic curve. The genus of $X$ can be defined as the dimension of the space of holomorphic $1$-forms on $X$, which in turn can be defined either …