# Questions tagged [analyticity]

A function is analytic if it has a converging power series expansion. Please also use one of (real-analysis) or (complex-analysis) to specify real or complex analyticity: though the definition is the same, the properties of functions analytic over real numbers are quite different from the properties of functions over the complex numbers.

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### Why is a polynomial in $z,\bar z$ analytic iff it does not involve monomials $z^i\bar z^j$ with $j\ge1$?

I have encountered some note on complex analysis: In particular, given a polynomial in the real variables x and y , with complex coefficients, these properties tell us that the polynomial is analytic ...
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### When is the zero set of a multivariate $p$-adic power series algebraic?

Let $f = f(z_1, \dots, z_n)$ be a power series in $n$ variables with coefficients in the $p$-adic integers $\mathbb{Z}_p$. Let $g(z_1, \dots, z_n) = f(pz_1, \dots, pz_n)$, so that $g$ converges on all ...
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### Real analytic manifold with curvature

Suppose $(M,g)$ is a real analytic Riemannian manifold (that is with coordinate charts that are real analytic functions with analytic inverses). Can we conclude that the manifold is always flat? The ...
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### Question about the "nowhere analyticity" of the Fabius function

As you could see in Wikipedia the Fabius function is a known "example of an infinitely differentiable function that is nowhere analytic". In the following answer where it is explained how is ...
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### Some tricky path integrals without residue theory

I am trying to work out the following path integrals without residue theory and have a really hard time doing so: \begin{align*} \int_{\partial K_2(0)}\frac{e^{iz^2}-1}{z^2}dz \...
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### curve selection lemma

$\quad$I am recently reading Proof of the Gradient Conjecture of R. Thom by Kurdyka, Mostowski, and Parusinski and I have a question about how to apply the curve selection lemma (CSL). Curve Selection ...
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### How do I find the principal branch?

I'm trying to find the domain on which a function is analytic, specifically, $\text{Log}\left(\frac{1}{z}+i\right)$. Would I need to find $\text{Log}\left(\frac{1}{z}+i\right)=\ln(r)+i\theta$ to find ...
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### Is $z \mapsto \int f(x,z) K_z(x) dx$ analytic if $K_z(x)$ is a distribution analytic in the paramter $z$?

Let $K_z(x) \in \mathcal D'(\mathbb R)$ be a distribution kernel depending analytically on $z \in O \subset \mathbb C$, i.e. for $f \in \mathcal D(\mathbb R)$ fixed, $$\int K_z(x) f(x) dx$$ is ...
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### Does the zero locus of a real analytic function have a smooth point?

Let $F \colon \mathbb{R}^m \to \mathbb{R}^n$ be a real analytic function (i.e., each of its component functions is real analytic). Does the set $$Z = \{x \in \mathbb{R}^m : F(x) = 0\}$$ have a smooth ...
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### Is there a surjective $C^{\omega}$ map from the 2-sphere to the 2-torus?

Is there a surjective $C^{\omega}$ map from the 2-sphere to the 2-torus? More explicitly, is there a surjective real-analytic mapping $f \colon S^2 \to T^2$ where S^2 = \{x \in \mathbb{R}^3 : \|x\| =...