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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.
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) …
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 …
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 …
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.
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 …