## Analog of Newlander–Nirenberg theorem for real analytic manifolds

It is a theorem of Whitney that every closed \$C^{infty}\$-manifold admits a real analytic structure. Furthermore, by a theorem of …

## Analytic continuation of holomorphic functions

Well, in case of power series some criterions do exist. Roughly speaking, one can take the element \$\$sumlimits_{n=1}^{infty}c_nz^n,quad z < …

## Analytic function avoiding elements of the modular group

Concerning question 2 see Earle, Clifford J. On holomorphic families of pointed Riemann surfaces. Bull. Amer. Math. Soc. 79 (1973), …

## Analytic implicit function theorem

One possible reference is “Holomorphic functions of several variables: an introduction to the fundamental theory” by Ludger Kaup and Burchard …

## Analyzing the decay rate of Taylor series coefficients when high-order derivatives are intractable

Complex analysis can help. The rate of Taylor coefficients is determined by: a) the radius of convergence, which is equal …

## An attempt to generalize the previous inequality

This is known to be Carlson’s inequality from 1935 (for \$t_kgeq 0\$, and not all \$t_k\$ are \$0\$). The Swedish …

## Algebraic vs analytic normality

Over \$mathbf{C}\$, algebraic normalization and analytic normalization are equivalent concepts. See N. Kuhlmann: Die Normalisierung komplexer Räume, Math. Ann. 144 …

## Algebraic independence of shifts of the Riemann zeta function

Hmm, it was more difficult than I expected to leverage universality to establish the claim. But one can proceed by …

## A harmonic function

Yes, it can be found explicitly, though not in elementary functions but in terms of a combination of elementary and …

## A generalization of holomorphic functions

Simple counterexample: Suppose \$f(x,y) = u(x,y) + i v(x,y)\$ is a holomorphic function. Then it satisfies the Cauchy-Riemann equations begin{align} …