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Results tagged with cv.complex-variables
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user 5371
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
134
votes
Interpreting the Famous Five equation
$$e^{i\pi}=\lim\limits_{N\to\infty}\left(1 + \frac{iπ}{N}\right)^N$$
- 22.3k
10
votes
What is the naming reason of poles in complex analysis?
According to these pages, the earliest known appearance of the term pole might be in "Théorie des fonctions elliptiques" (1875, p. 15) by Briot and Bouquet:
Lorsqu'une fonction $u$ est holomorph …
14
votes
Must the set of lines through the origin on which a nonconstant entire function is bounded b...
The Mittag-Leffler function $E_{\alpha,1}$, $\alpha>0$, is bounded in the sector
$$\frac{\alpha\pi}{2}< \arg z<2\pi-\frac{\alpha\pi}{2}.$$
In particular, $e^z=E_{1,1}(z)$ is bounded in
$$\frac{\pi}{ …
- 22.3k
15
votes
Accepted
Analytic continuation of holomorphic functions
Well, in case of power series some criterions do exist. Roughly speaking, one can take the element
$$
\sum\limits_{n=1}^{\infty}c_nz^n,\quad z < 1,\label{1}\tag{$\ast$}$$
and consider an analytic func …
- 22.3k
13
votes
Does Riemann map depend continuously on the domain?
Yes. This is a classical result of the geometric function theory due to Carathéodory.
The theorem and a fairly straightforward proof can be found, for instance, in the Hurwitz-Courant Funktionentheo …
- 22.3k
2
votes
Accepted
Approximation by analytic functions
The answer to the modified question is given by Jackson-type theorems.
The classic book by N.I. Akhiezer which is quoted in the Wikipedia article contains a number of specialised results on optimal a …
- 22.3k
22
votes
Accepted
Ramanujan's eccentric Integral formula
This is one of those precious cases when Ramanujan himself provided (a sketch of) a proof. The identity was published in his paper "Some definite integrals" (Mess. Math. 44 (1915), pp. 10-18) togethe …
- 22.3k
9
votes
Accepted
Holomorphic function with a.e. vanishing radial boundary limits
I am not sure if this would qualify as 'easy' but the first example of such a function was constructed by Lusin. It can be found in N. Lusin, J. Priwaloff, Sur l'unicité et la multiplicité des fonctio …
- 22.3k
8
votes
Accepted
Dirichlet series expansion of an analytic function
A.F. Leont'ev continued to work on general Dirichlet series well into 1980s (until his death in 1987). Actually, he published three monographs on the subject from 1976 to 1983! He made a short summary …
- 22.3k
8
votes
Reference for complex analysis jargon
I would say these concepts rather belong to the field of potential theory. You will find most of the definitions and a fairly advanced treatment of the subject in Logarithmic Potentials with External …
- 22.3k
14
votes
If the Riemann Hypothesis fails, must it fail infinitely often?
To quote from "Problems of the Millennium: the Riemann Hypothesis" by Bombieri:
It is known that hypothetical exceptions to the Riemann hypothesis must be rare if we move away from the line $\Re(s) = …
- 22.3k
22
votes
Accepted
When I can safely assume that a function is a Laplace transform of other function?
The answer depends on the class of functions $\phi(t):(0,\infty)\to\mathbb R$ where you want to define the Laplace transform. A standard assumption is that
$$e^{-ct}\phi(t)\in L^2(0,\infty)\tag{1}\lab …
- 22.3k