Mathematics Stack Exchange
Mathematics Stack Exchange

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.

If $U$ is a subset of $\mathbb{C}$, $f$ is analytic at $x_0$ if there exists a series $$ \sum_{j=0}^\infty a_j (z-z_0)^j $$ that converges to $f$ at a neighbourhood of $z_0$. In complex analysis, analyticity is equivalent to holomorphy.

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The distinction between infinitely differentiable function and real analytic function

I have known that all the real analytic functions are infinitely differentiable. On the other hand, I know that there exists a function that is infinitely differentiable but not real analytic. For example, $$f(x) = \begin{cases} \exp(-1/x), &…
Juntao Huang
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Sequence of functions which approximate analytic function

Consider the function $f(z)=\frac{1}{z}$ on the annulus $A=\{z \in \mathbb{C} : \frac{1}{2} \lt{\vert{z}\vert} \lt2\}$. Which of the following are true? There is a sequence $\{P_n(z)\}_n$ of polynomials that approximate $f(z)$ uniformly on compact…
dipali mali
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Suppose f and g are analytic on a domain G

Suppose f and g are analytic on a domain G. If f and g are non-constant, then for any b in G, there exists a punctured disk D'(b,R) of radius R>0 such that f(z)g(z) is not equal to g(z)-f(z) + 1 for all z in D'(b,R) IS this sentence incorrect? Thank…
aD1234567890
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Are analytic functions injective?

Let $f$ be analytic on the whole all of $\mathbb C$. Assume that $\mathrm{Re}\, f \ge 0$. What can we say about $f$? I'm thinking $f$ has got to be constant, since otherwise it would map the entire complex plane onto the positive real half-plane. I…
Benjamin Lindqvist
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Is composition of analytic functions itself analytic?

Is composition of analytic functions itself analytic? Is there a proof that, say, $$f(x)=e^{\frac{x^2+1}{x^2-1}}$$ analytic?
Anixx
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existence of an analytic function in unit disk

Does there exists an analytic function $f$ in unit disk such that $f(-\frac{1}{2})=3$, $f(n^{-2})=5$ for $n\ge 2$. i am not able to solve any help will be appreciated.
user148789
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Inequality regarding entire function

Let $f$ be entire function. Must there exists $R>0$; such that $|f(z)| \leq |f'(z)|$ for all $|z|>R$ ,OR $|f'(z)| \leq |f(z)|$ for all $|z|>R$ ?
math123
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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 Lemma: Let $X\subset\mathbb{R}^n$ be a…
polyroot
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If both $f(z)$ and $−\overline{f(z)}$ are analytic, what can you say about $f(z)$? Prove your claim.

I am struggling with this question. If both $f(z)$ and its negative conjugate $-\overline{f(z)}$ are analytic, what can you say about $f(z)$? Prove your claim. I know that the sum on difference of two analytic functions is analytic but I can't see…
Elli
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Does this proof contain a contradiction?

I have made the following proof and I am asking if there is anything wrong in my steps: Let $f:\mathbb{R}\to\mathbb{R}$ be a real analytic function with infinitely many zeros. Let $a\neq 0$ be a real number. We know that the set $f^{-1}(a)$ is…
Safwane
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Analytic function problem

Let $I \subseteq \Bbb R$ be non-empty open interval. Let $f : I \rightarrow \Bbb R$ be a real analytic function. Let $y_0 \in I$ be a point such that $f^{\prime}(y_0) \neq 0$ (a) Show that there is an open interval $J \subseteq I$ containging $y_0$…
fivestar
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Products of power series

consider the identity $$\frac{e^{-x}}{1-x}=\sum_{n=0}^{\infty}c_nx^n$$ Show that for each $n\ge0$ $$\sum_{k=0}^{n}\frac{c_k}{(n-k)!}=1$$ My trial : By cauchy product,…
fivestar
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What will be $f^{'}(0)$ and $f(\dfrac{1}{3})$?

let $f:D=\{z\in \mathbb C:|z|<1\} \to \overline D$ with $f(0)=0$ be a holomorphic function. What will be $f^{'}(0)$ and $f(\dfrac{1}{3})$? My try:By cauchy integral formula : $f^{'}(0)=\dfrac{1}{2\pi i}\int_\gamma \dfrac{f(z)}{z^2}dz$ where $\gamma$…
Learnmore
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Are the coefficients of power series expansion for a real analytic function bounded?

Are the coefficients of power series expansion for a real analytic function bounded? $f(x)=\sum_{n=0}^{\infty}a_n (x-x_0)^n$ We have a sequence $\{a_n\}, n=0,\cdots,\infty$. Is this sequence bounded? Thanks.
Sophia Tang
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Zero of non analytic function

Let a function $L=L(z)$ be analytic, for $\mathrm{Re}\, z>0$, and be singular at $0$, however, $L(0)=c$ be finite. Let also $L'(0)$ be finite as well, however, $L'(0)\neq 0$. For example, $$ L(z)=\int_0^\infty \frac{e^{-z t}}{1+t^3}dt. $$ May I…
Dmitri
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