# Questions tagged [analytic-functions]

For questions about analytic functions, which are real or complex functions locally given by a convergent power series.

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### For continuous $f(x)$, if $\lim_{x\to 0} f(x)/p(x)=0$ for every nonzero polynomial $p(x)$, then is the same true for nonzero analytic $p(x)$?

Let $f(x)$ be a continuous real function on a neighborhood of 0 in $\mathbb{R}$. Suppose $f(x)>0$ for $x\neq0$, and suppose $\lim_{x\to0} f(x)/p(x)$ for every nonzero polynomial $p(x)$. My question ...
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### Are Characteristic Functions Analytic?

Let $X$ be a real random variable defined on a probability space $(\Omega, F, P)$. Define its characteristic function $\phi: \mathbb{R} \to \mathbb{C}$ by $\phi(t) = \mathbb{E}[e^{itX}]$ for every ...
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### Is there any simple set of properties that uniquely characterizes differentiation in the space of complex functions?

The transformation of differentiation is a linear operator over the vector space of entire functions (call this space $\mathbb{C}^E.$) Is there any simple set of properties that uniquely determines ...
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### What can one tell about the derivatives of a function which intersects a polynomial of degree $n$ at no more than $n+2$ points?

Consider an infinitely differentiable function $f:\Bbb R \to \Bbb R$ and define $\phi(n), n\in \Bbb N_0$ the maximum number of roots that the function $f(x)-p(x)$ can have for any polynomial of degree ...
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### If any $n$-degree polynomial intersects $f$ at no more than $n+1$ then $f$ has all its derivatives positive.

Conjecture. An infinitely differentiable function $f:\Bbb R \to \Bbb R$ at some $x\in \Bbb R$ has all its derivatives nonzero with identical or alternative sign if and only for any polynomial $p$ the ...
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### Is the real part of a complex analytic function, real analytic?

Recently in Calculus 1 we were introduced to the concept of analytic functions (to be more exact, real analytic functions). At the same time I was familiar with the concept of complex analytic ...
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### Modulus squared of derivative of analytic function [closed]

In my lecturers notes it is mentioned: $(u_{x})^2 + (v_{x})^2 = |\frac{dw}{dz}|^2,$ where $w = u(x, y) + iv(x, y), \quad z = x + iy.$ But this isn't immediately obvious to me, if anyone could offer ...
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### For which sequence do all associated power series converge? (describe the subset of $\Bbb R^\Bbb N$ "homeomorphic" to the analytic functions)

I am investigating a certain property of real smooth functions (though it is easily extended to complex analytic functions) which requires me to define a topology on $C^\infty(\Bbb R)$ that "...
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