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 is quite different from the properties ...

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11 views

Singularities of quotient of polynomials where the degree of the denominator $\ge$ the degree of the numerator $+2$.

Let the degrees of the polynomials $$P(z)=a_0+a_1 z+a_2 z^2+\cdots +a_n z^n \; (a_n \neq 0)$$ and $$Q(z)=b_0+b_1 z+b_2 z^2+\cdots +b_m z^m \; (b_m\neq 0)$$ be such that $m \ge n+2.$ Show that if ...
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53 views

Is $f(a+b\epsilon)=f(a)+b\epsilon f'(a)$ true for non-analytic smooth functions of dual argument

Does $f(a+b\epsilon)=f(a)+b\epsilon f'(a)$ remain true for non-analytic smooth functions of dual number argument? Where $ \epsilon^2=0$, $\epsilon \neq 0$ and $a,b \in \mathbb R$. I found proofs ...
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37 views

How does analytic continuation lets us extend functions to the complex plane?

I'm trying to understand analytic continuation and I noticed on wolfram that it allows the natural extension of the definition trigonometric, exponential, logarithmic, power, and hyperbolic ...
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12 views

Jensen's formula for a cosine function with disk of radius $\gamma_n$ the imaginary part of the nth non-trivial zero of the Riemann zeta function

I would like that you tell me the computations to get the deduced result from Jensen's formula (is the first formula in the section The statement of this Wikipedia's Page) in the case that our ...
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22 views

Does this transformation and reparametrization preserve non-analyticity of smooth functions

A $C^{\infty}$ function $f : \mathbb{R} \to \mathbb{R}_+$ is non-analytic for some $x_0 \in \mathbb{R}$. Can we conclude that $x\mapsto\log(f(e^x))$ is non-analytic at $x = \log(x_0)$?
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14 views

choosing coefficients of a real analytic function so that it oscillates on $(-\infty,0)$

Fix a sequence $(a_n)_{n=0}^\infty$ of positive real numbers satisfying $\sqrt[n]{a_n}\to 0$, so that the rule $F(x)=\sum_{i=0}^\infty a_ix^i$ defines a function $F:\mathbb{R}\to\mathbb{R}$. ...
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25 views

Analyzing behaviour of a complex function with certain properties

Let $f$ be a complex function with the following properties: At the origin, $z = 0$, $f$ is analytic and representable by a power series with finite radius of convergence $\rho$. At the (real) ...
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27 views

Is being real analytic at a point equivalent of matching the Taylor Series around that point?

Let $U \subset \mathbb{R}$ be an open interval and let $x_0 \in U$. Let $f: U \to \mathbb{R}$ be defined by a power series around $x_0$ with radius of convergence $R > 0$, $$f(x) = \sum_{n=0}^{\...
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Fibonorial of a fractional or complex argument

Let $F(n)$ denote the $n^{\text{th}}$ Fibonacci number$^{[1]}$$\!^{[2]}$$\!^{[3]}$. The Fibonacci numbers have a natural generalization to an analytic function of a complex argument: $$F(z)=\left(\phi^...
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Does a analytic joint distribution necessarily have continuous marginals?

Although the question actually popped up in a course about evolution equations, it seemed most natural to ask this in the context of joint distributions. Namely: Suppose $X$ and $Y$ are two ...
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27 views

Analyticity Proof

Suppose $f$ is analytic on $(a, b)$. Prove that if $(c, d)$ is a subinterval of $(a, b)$ and $f(x) = 0$ for all $x$ in $(c, d)$, then $f(x) = 0$ for all $x$ in $(a, b)$.
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26 views

Does infinitely differentiable imply analytic? [duplicate]

I know that analytic implies infinitely differentiable, but is the converse always true as well?
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97 views

Proving analyticity of an integral function over $\mathbb{R}^{n}$

Let $U\subsetneqq\mathbb{R}^{n}$ be open, $\varepsilon>0$ and consider the function $$f_{\varepsilon}(x)=\frac{\pi^{-\frac{n}{2}}}{\varepsilon^{n}}\int_{U}\exp\left\{-\left\|\frac{x-y}{\varepsilon}\...
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36 views

Uniformly bounded sequence of analytic functions in the unit disk

Suppose $\{f_n\}$ is sequence of analytic functions that is uniformly bounded in the open unit disk and for every positive integer $k$, $f_n(\frac{1}{1+k})\to 0$ pointwise. Then, $\{f_n\}$ converges ...
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79 views

A smooth nowhere analytic function such that all derivatives are monotone

Related questions that might provide some context: (1) (2) (3) (4) Let's restrict our attention to real-values functions on an open unit interval $f:(0,1)\to\mathbb R$. There are examples $\!^{[1]}$...
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28 views

Interpolating discrete data with completely monotone analytic functions

Suppose we have a positive integer $n$ and a finite list of real numbers $\{a_1,\,a_2,\,\dots,\,a_n\}$. We want to find a real-analytic function $f:[1,n]\to\mathbb R$ such that $f(m)=a_m$ for all $m\...
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35 views

analyticity of an implicitly defined function

Let $F:A\subseteq\mathbb{R}^2\to\mathbb{R}$ be a function, where $A$ is open in $\mathbb{R}^2$ and $F\in\mathcal{C}^k(A)$ (we take $\mathbb{R}^2$ and $\mathbb{R}$ for the sake of simplicity), and let $...
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1answer
30 views

Analyticity of roots of a polynomial in terms of coefficients

Suppose that $f(z,w)$ is a non-constant polynomial in $z,w$ with coefficients in $\mathbb{C}$. Fix $z$, we define $p(w)=f(z,w)$. From Liouville's theorem, we know that $p(w)=0$ is solvable for $w$, ...
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20 views

Conformal map from doubly slit plane to the open unit disk.

As stated in the title, what is the starting point in finding a conformal map between doubly-slit domain to the open unit disk? I know how to deal with a single-slit domains, but have trouble trying ...
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35 views

Can we Relate Radius of Convergence of Taylor Series and Asymptotic Rate of Growth?

I still need to be disabused of the belief that there is some simple connection between the finiteness of the radius of convergence and the asymptotic rate of growth. 1. Can we develop any ...
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10 views

real analytic extension of a composition of real analytic functions

Let $H \colon \mathbb{R}^n \to \mathbb{R}^k$ be a real analytic function, as well as $F \colon \mathbb{R}^n \to \mathbb{R}^l$ and $G \colon F(U) \to \mathbb{R}^k$ for some open set $U \in \mathbb{R}^...
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32 views

Find an analytic continuation

Let $f(z)=\sum_{j=0}^{\infty}z^j$ for $|z|<1$. For what values of $\alpha$ ($|\alpha|<1$) does the Taylor expansion of f(z) about $z=\alpha$ yield a direct analytic continuaton of f(z) to a disk ...
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29 views

Analyticity and differentiability of complex functions

I understand what analytic functions are and what differentiability of a complex function means but I have been reading "advanced engineering mathematics by kreyszig" and it says that the concept of ...
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15 views

Why is $\{(1/(n+1),n)\mid n\in\mathbb{N}\}$ a subanalytic set?

I have a question about subanalytic sets. Why is $\{(1/(n+1),n)\mid n\in\mathbb{N}\}$ a subanalytic set but its projection onto $\mathbf{R}\times \{0\}$ is not? Thanks in advance,
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1answer
55 views

Finding residues at a point $a$ where $a$ is a pole.

I am faced with the following problem: (a) Find $\displaystyle res_{a} \frac{\varphi(z)}{(z-a)^{n}}$ where $\varphi$ is a given function analytic at $a$, $\varphi(a) \neq 0$, and $n$ is a positive ...
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22 views

Finding an analytic function

enter image description here I cannot find any such function. Also, why would a function that is analytic at 0 following these criteria not be analytic on (-2,0). Thanks in advance for your help.
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37 views

Identity theorem for forms (on analytic manifolds)

It is well-known that for holomorphic functions the Identity theorem holds: if two holomorphic functions agree on an open subset, they agree everywhere (assuming the manifold connected). I would ...
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68 views

an analytic function being zero

Let $f$ be an analytic function defined on the unit disc $D=\{z:|z|<1\}$. If $|f(z)|\leq 1-|z|$ for all $z\in D$ then show that $f$ is a zero function on $D$. Please give only hints.
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$f$ has pole of order $m$ and $g$ has a pole of order $n$ at $z_{0}$, show $f+g$ has isolated singular point there

I am faced with the following problem: Suppose $f(z)$ and $g(z)$ have poles of order $m$ and $n$ respectively, at a point $z_{0} \in \mathbb{C}$ with $m \neq n.$ Show that $z_{0}$ is an isolated ...
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31 views

Let $f$ be analytic in $D(0,2)$. Assume that for all $n∈\mathbb{N}$ $\int_{ |z|=1} {f(z)\over(n+1)z−1}dz=0$. Prove that $f(z)=0$ for all $z∈D(0,2)$.

Let $f$ be analytic in $D(0,2)$. Assume that for all $n∈\mathbb{N}$ $\int_{ |z|=1} {f(z)\over(n+1)z−1}dz=0$. Prove that $f(z)=0$ for all $z∈D(0,2)$. I'm thinking about a contradiction proof. Assuming ...
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60 views

Suppose $f$ is analytic and $f(a) = f(b) = 0$. Show that $|f(z)| ≤ |{z − a \over 1 − z\bar{a}}| · |{z − b \over 1 − z\bar{b}}|$.

Suppose $f$ is analytic from $D(0, 1)$ to $D(0, 1)$ and $f(a) = f(b) = 0$ for two different numbers $a, b$ in $D(0, 1)$. Show that $\left\vert f(z) \right\vert ≤ \left\vert{z − a \over 1 − z\bar{a} }\...
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Prove that $f$ is analytic and derivatives of all order $f_n^{(k)}$ converge to $f^{ (k)}$ uniformly on any compact subset of $G$.

Suppose $f_n$ is analytic in some region $G$ and suppose $f_n$ converges to $f$ uniformly on any compact subset of $G$. Prove that $f$ is analytic and derivatives of all order $f_n^{(k)}$ converge to $...
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1answer
14 views

Domain of holomorphicity of $f(z)=\frac{1}{e^z+1}$

Domain of holomorphicity of $$f(z)=\frac{1}{e^z+1}$$ Is this simply everywhere, because there is no $\bar z$ dependence?
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67 views

$f(|z|)$ is not an analytic function

Let $f: [0,\infty)\rightarrow \mathbb{C}$ is a non constant function. Define $g:\mathbb{C}\rightarrow\mathbb{C}$ by $g(z)=f(|z|)$. Prove that $g(z)$ is not holomorphic. So, I need to find a point $...
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59 views

Powers of a function being analytic [duplicate]

Question is as follows : Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is continuous such that $f^3,f^4$ are analytic in $\mathbb{C}$ then prove that $f$ is analytic in $\mathbb{C}$.. Choose $...
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1answer
49 views

Entire function bounded sequence

Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be an entire function such that $|f(1/n)|\leq 1/n^{3/2}$ for all $n\in \mathbb{N}$ then show that $\{n^2f(1/n)\}$ is bounded. To show that $\{n^2f(1/n)\}$ is ...
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66 views

Function analytic in each variable does not imply jointly analytic

I have heard that a function $f: \mathbb R^2 \to \mathbb R$ can be analytic in each variable (i.e. $f(x,y_0) = \sum_{n=0}^{\infty} a_n x^n, \forall x \in \mathbb R$, and the same for $y$) without ...
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Is this function analytic: $F(x+iy)=\frac1{\pi}\int_{\mathbb R}\frac y{(x-t)^2+y^2}\,f(t)\,dt$.?

Let $f:\mathbb R\to\mathbb R$ such that $f(t)=0$ if $|t|\leq 1$ and $f(t)=|t|^\lambda$ if $|t|>1$. Here $\lambda<0$ is a constant. We consider the following function on $\mathbb C_+=\{x+iy:\;x,...
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Let $f$ be analytic on the unit disk $D = \{z : |z| ≤ 1\}$ and suppose $Im(f(z)) > 0$ for $z ∈ D$ and $f(0) = i$. Prove that $|f ' (0)| ≤ 1$. [closed]

Let $f$ be analytic on the unit disk $D = \{z : |z| ≤ 1\}$ and suppose $Im(f(z)) > 0$ for $z ∈ D$ and $f(0) = i$. Prove that $|f ' (0)| ≤ 1$. For what functions do we have equality? I'm not sure ...
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1answer
33 views

Analytic, non- constant complex function has finitely many zeros inside the disk $D(0, R)$ for all $R > 0$

I'm learning about complex analysis and need help with the following problem: Let $f: \mathbb{C} \to \mathbb{C}$ be analytic and non-constant. Show that for every $R > 0$, the complex function $...
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1answer
42 views

Application of Schawarz lemma??

Let $f$ be an analytic function defined on the unit disc $D=\{z:|z|<1$. If $|f(z)|\leq 1-|z|$ for all $z\in D$ then show that $f$ is a zero function on $D$. We have $|f(z)|\leq 1-|z|$ for all $z$ ...
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15 views

Mobius transformations

Suppose f is a continuous function on the extended complex plane which is analytic except possibly at one point and maps lines and circles to lines and circles. Does it follow that f is necessarily a ...
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1answer
24 views

Interior Uniqueness: Does there exist an analytic function on a neighborhood of $z=0$ that satisfies the following?

I am faced with the following problem: Does there exist a function that is analytic on a neighborhood of $z=0$ and satisfies the following condition for every positive $n$: (a) $f(1/n)=f(-1/n)...
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0answers
26 views

Local Uniform Convergence and Composition

I've been sitting down can't quite tell if this is true or not, but I suspect that it should be. Edit: Suppose that $\Omega$ is a open, connected subset of $\mathbb{C}$, and suppose that $(f_n) \...
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38 views

function analytic in the entire complex plane is constant

Let the function $f$ be analytic in the entire complex plane and suppose that $\frac{f(z)}{z}\rightarrow 0$ as $|z|\rightarrow \infty$. Prove that $f$ is a constant. As $\frac{f(z)}{z}\rightarrow 0$ ...
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205 views

Laurent expansion of $\frac{1}{\sin z}$

Question is a fully solved exercise in Gamelin's complex analysis. Exercise : Consider the Laurent series expansion for $\frac{1}{\sin z}$ that converges on the circle $\{|z|=4\}$. Find the ...
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2answers
48 views

Why is this point analytic?

Suppose you had the function $$ p(x) = \frac{\sin(x)}{x} $$ I know, from other material online, that this point is analytic at the point $x=0$. However, my understanding was that a point of a function ...
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0answers
30 views

How to prove this function is entire?

Given a function which Fourier coefficient decay fast as $k^{-k}$, for example \begin{align} f(x):= \sum_{k=1}^\infty \frac{1}{k^k} \exp(2\pi \, i\,k\,x) . \end{align} How can we prove this function ...
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13 views

How is it possible to find all singular (non-analytic) points of a differential equation?

Suppose we had a second order differential equation of the form $$ y'' + p(x)y' + q(x)y = 0 $$ I know that a point $x=x_{0}$ is said to be ordinary if $p(x_{0})$ and $q(x_{0})$ can be expressed as an ...
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2answers
38 views

Laurent expansion

Question is to write Laurent expansion of $f(z)=\dfrac{1}{(z-2)(z-1)}$ in the annulus $1<|z|<2$ based at $z=0$ I am aware of the method of partial fractions and writing expansions for both $\...