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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 …

I would be interested to learn if the following generalization of the classical Looman-Menchoff theorem is true. Assume that the function $f=u+iv$, defined on a domain $D\subset\mathbb{C}$, is such t …

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 …

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 …

Problem. Let $\{\lambda_n\}_{n\in\mathbb N}$ be a sequence of complex numbers . Let's call a family of exponential functions $\{\exp (\lambda_n s)\}_{n\in\mathbb N}$ $F$-independent (where $F$ is e …

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 …

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 …

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 …

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) = …

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 …

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 …

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}{ …

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 …

$$e^{i\pi}=\lim\limits_{N\to\infty}\left(1 + \frac{iπ}{N}\right)^N$$