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Results tagged with cv.complex-variables
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user 22277
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
3
votes
Poisson inequality for subharmonic functions
The book Potential Theory in the Complex Plane by Thomas Ransford has a proof of this fact on page 35. I learned potential theory in two dimensions from this book.
7
votes
Is anything known about the series $\sum_{n=0}^{\infty} x^{\sqrt{n}} $?
I claim that the function $G$ satisfies a few functional equations, but in order to formulate our functional equation, we need to extend $G$ to a much larger domain. One often has to extend a function …
3
votes
Any closed form for series like $F(x)=\sum\limits_{p=2}^{\infty}x^p,$ where $p$ is prime?
Functions with a natural boundary tend to satisfy functional equations, and they are sometimes the unique solutions to those functional equations. While a system of functional equations is far from be …
204
votes
Accepted
Why do roots of polynomials tend to have absolute value close to 1?
Let me give an informal explanation using what little I know about complex analysis.
Suppose that $p(z)=a_{0}+\dotsm+a_{n}z^{n}$ is a polynomial with random complex coefficients and suppose that $p(z) …
4
votes
Accepted
Has anyone characterized the zeroes of the Bell numbers?
The function $B(z)$ is an example of an almost periodic function. The zeroes of an almost periodic function that is holomorphic on some strip are also almost periodic, so such a function either has no …