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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1answer
50 views

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: ...
3
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0answers
26 views

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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1answer
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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1answer
25 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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1answer
93 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 ...
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1answer
34 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 ...
6
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1answer
72 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 ...
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0answers
27 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 ...
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1answer
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 ...
0
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1answer
28 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
17 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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2answers
33 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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0answers
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 ...
0
votes
0answers
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 ...
1
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1answer
28 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 ...
0
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0answers
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,
2
votes
1answer
54 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 ...
0
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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.
2
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1answer
35 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 ...
3
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2answers
67 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.
2
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0answers
36 views

$f$ 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 ...
0
votes
1answer
30 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 ...
0
votes
1answer
59 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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0answers
23 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 ...
0
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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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3answers
64 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
votes
2answers
57 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
vote
1answer
43 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
votes
2answers
62 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 ...
0
votes
1answer
17 views

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 ...
1
vote
2answers
70 views

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 ...
1
vote
1answer
28 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 ...
1
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1answer
40 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 $z$ ...
0
votes
0answers
12 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 ...
0
votes
1answer
22 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) ...
0
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0answers
25 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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0answers
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 $\frac{f(z)}{z}\rightarrow 0$ ...
5
votes
0answers
185 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 ...
0
votes
2answers
47 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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0answers
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
37 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 ...
2
votes
1answer
54 views

Power of a function is analytic [duplicate]

Question is : Show that if $f(z)$ is continuous function on a domain $D$ such that $f(z)^N$ is analytic on $D$ for some integer $N$ then $f(z)$ is analytic on $D$.. For some time i was wondering ...
0
votes
1answer
15 views

The existence of the roots of an holomorphic function on an open connected domain

Let $U$ be an open connected domain and $D$ be an open disk such that the closure of $D$ is a subset of $U$. Suppose $f\in H(U)$, i.e., $f$ is holomorphic in $U$, and that $f$ is not constant. Show ...
0
votes
1answer
19 views

Use result that composition of analytic functions is harmonic to find harmonic conjugate of $e^{-2xy}\sin (x^{2}-y^{2})$

Using this result, I need to find a harmonic conjugate for $e^{-2xy}\sin(x^{2}-y^{2})$. In that result, I'm supposing that $s = e^{-2xy}$ and $t = \sin(x^{2}-y^{2})$, but I really don't know how to ...
1
vote
1answer
38 views

Evaluate $\int_{|z|=1}\frac{|dz|}{|z-a|^2}$ using Cauchy's Theorem.

Question: Evaluate $\int_{|z|=1}\frac{|dz|}{|z-a|^2}$ using Cauchy's Theorem. My attempt: So, Cauchy's theorem for derivatives tells us that if $f$ is holomorphic in an open set $\Omega$, and $D$ is ...
2
votes
1answer
28 views

Let the function $f$ be analytic in $C$, real valued on $R$, and $\Im f(z) > 0$ in the upper half-plane. Prove that $f'(x) > 0$ for $x \in R$.

Question: Let the function $f$ be analytic in the entire complex plane, real valued on the real axis, and of positive imaginary part in the upper half-plane. Prove that $f'(x) > 0$ for $x \in R$. ...
1
vote
1answer
14 views

Analytic function on the annulus $A = \{z : 1 \le |z| \le 4\}$ onto $B = \{z : 1 \le |z| \le 2\}$ s.t. $C_1 \to C_1$, $C_4 \to C_2$?

Question: Does there exist an analytic function mapping $A = \{z : 1 \le |z| \le 4\}$ onto $B = \{z : 1 \le |z| \le 2\}$ and taking $C_1 \to C_1$, $C_4 \to C_2$, where $C_r$ is the circle of radius ...
2
votes
0answers
46 views

A single analytic function that can approximate all others

The problem in it's entirety is this: Given some simply connected domain $U$ then show that there is a function $g\in\mathcal{O}(\mathbb{C})$ such that for any given $f$ there exists a sequence ...
0
votes
1answer
22 views

Let $f(z) = u(x,y) + iv(x,y)$ be analytic in $\Omega$, suppose that $v(x,y) = e^{-y}(y\cos x -x \sin x)$, find $f(z)$.

Question: a) Let $f(z) = u(x,y) + iv(x,y)$ be analytic in $\Omega$, suppose that $v(x,y) = e^{-y}(y\cos x -x \sin x)$. Find $f(z)$. b) Let $f(z), g(z)$ be analytic in an open, connected domain ...