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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Linear combination of a real-valued function and its inverse is analytic Implies the real-valued function is analytic.

If $u$ is a real-valued function on a disc $\Delta_R$ such that $u^{-1}+iu$ is analytic on $\Delta_R$, then does this imply that $u$ is analytic on $\Delta_R$? I am actually trying to prove some ...
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Analytic Functions

Prove or give a counter-example: If $f_j(j=1,2,...,n)$ is analytic on the domain $D$ such that $\sum_{j=1}^n |f_j(z)|^2$ is constant on $D$. Then each $f_j$ is a constant function. Inputs: We know ...
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18 views

Symmetry in Analytic Continuation of $\sum_{n=0}^{\infty} e^{-x E_n}$

Suppose we have the following function: $$F(x)=\sum_{n=0}^{\infty} e^{-x E_n}$$ Where $E_n$ is a positive monotonically increasing sequence, bounded from below. Is there a general condition on $E_n$ ...
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29 views

Analytic function zero in the given disk

I need to show that f(z)=0 for all z \in D(0,2). From the analyticity of f in D(o,2), I know by Cauchy's theorem it's integral in |z|<2 is zeros. And clearly the integrand has a pole at 1/(n+1) ...
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35 views

$(B_t)_{t\ge 0}$ be Brownian motion. Then $\xi \mapsto \mathbb E e^{i\xi B_t}$ is an analytic function. [closed]

Let $(B_t)_{t\ge 0}$ be a one-dimensional Brownian motion. Then $\xi \mapsto \mathbb E e^{i\xi B_t} \; \text{for all} \; t\ge 0, \xi \in \mathbb{R}$ is an analytic function. A more general question ...
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44 views

Describe all real-valued functions which are analytic on $\mathbb{C}$

This is a homework question, so if I am wrong please do not explicitly give me the answer. Question: Describe all real-valued functions which are analytic on $\mathbb{C}$. My Answer: Given that we ...
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28 views

Is $\exp \left(-\sum_{i=1}^d \frac{(x_i - y_i)^2}{s_i^2} \right) $ analytic in $\mathbf{x}$ on $\mathbb{R}^d$?

I would like to know whether the following statement is true. Conjecture: For all $\mathbf{y} \in \mathbb{R}^d$, $s_1>0,\ldots,s_d>0$, $f_{\mathbf{y}}(\mathbf{x}) := \exp \left(-\sum_{i=1}^d \...
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40 views

Proving directly (without integration) that differentiable implies analytic for one complex variable

From Titchmarsh, Theory of Functions, 2nd ed 1939: [His definition of an analytic function appears to be that $f(z)$ is analytic at $z_0$ if $\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}$ exists (...
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33 views

Help to understand this step of the proof of holomorphic functions are analytic

I'm trying to understand the proof of the analyticity of holomorphic functions. The step I don't understand is when one interchanges the series and the integral. In every source I have read, it says ...
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36 views

Isn't this condition for $f$ to have a pole of order $k$ redundant?

From Complex Analysis by Bak and Newman (p. 118): If $f$ is analytic in a deleted neighborhood of $z_0$ and if there exists a positive integer $k$ such that $$\lim_{z→z_0} (z − z_0)^k f (z) \neq 0 ...
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58 views

Showing Complex Function is Constant

I am preparing for qualifying exams, and this is a question from the Penn State Qualifying Exam for Fall 2015. It is stated as follows Let $\epsilon > 0$ and let $f$ be holomorphic (analytic) on ...
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19 views

Proof of analyticity in Lewy's Example:

I was reading: https://people.maths.ox.ac.uk/trefethen/pdectb/lewy2.pdf where it is stated: The Lewy Operator on a function $f(x,y,t) : x,y,t \in \mathbb{C}$ is given by $$ \frac{\partial f }{\...
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231 views

Sum and product of analytic functions that is not analytic

The function $$f(x) = \frac{2 + \cos x}{3} (2π - x) + \sin x$$ is the sum/product of analytic functions ($\cos(x)$,$\sin(x)$, linear), but all it's derivatives at $2\pi$ are $0$ ($f^n(2\pi)=0$). I ...
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14 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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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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1answer
38 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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13 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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1answer
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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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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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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98 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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39 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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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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31 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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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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1answer
21 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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40 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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11 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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33 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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1answer
32 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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0answers
16 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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1answer
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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1answer
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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70 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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1answer
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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61 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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24 views

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?
2
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3answers
68 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 $...
3
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2answers
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 $...
1
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1answer
52 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 ...
5
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2answers
68 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 ...