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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28 views

Proving that $f(z)\neq \frac{z}{z+1}$ in $D_1(0)$

Suppose $f$ is analytic in $D_r(0)$ for some $r>1$. I want to prove that $f(z)\neq \frac{z}{z+1}$ in $D_1(0)$. This is how I tried to prove this. Assume $f(z)= \frac{z}{z+1}$ in $D_1(0)$. Now ...
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0answers
43 views

Showing that there exist $C \in \mathbb{C}$ such that $g(z)=C \sin(z)$ if $g$ holomorphic & $|g(z)|\leq A|\sin(z)|$ ($A\in\mathbb{R}$)? [duplicate]

I'm trying to manipulate the sine function is some complex analysis problems (I need practice) and I've encountered two slight darker points: First, I don't understand how it can be possible (I read ...
4
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1answer
67 views

Laurent series, integral over the annulus, radii

We are given $$f = \sum_{n= - \infty} ^{\infty} a_n (z-z_0)^n \in \mathcal{O} ( \text{ann} (z_0, r, R)), \ \ 0<r<R< \infty. $$ Prove that $$\frac{1}{\pi} \int _{ann (z_0, r, R)} |f(z)|^2 d ...
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1answer
37 views

Do real-analytic functions always extend uniquely to complex-analytic functions on $\mathbb{C}$?

A function $f(x)$ is an real function and analytic in an open interval of $x$-axis or the whole $x$-axis. Is there only unique way to analytically extend it to the whole complex plane? I know ...
3
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2answers
66 views

Which entire functions satisfy $\,\lvert\,f(z)\rvert \leq \lvert z\rvert^k$?

Which holomorhpic functions satisfy $\,\lvert\,f(z)\rvert \leq \lvert z\rvert^k$ on $\mathbb{C}$? So I've shown that $\,\lvert\, f(z)\rvert \leq \lvert z\rvert ^k \implies f(z)$ is a polynomial of ...
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2answers
72 views

Prove that if $\,\lvert\, f(\,f(z))\rvert>r,\,$ then $f$ is constant

Let $r>0$. Prove that if $f$ is holomorphic on a whole complex plane and $|f(f(z))|>r$ for all $z\in\mathbb{C}$, then $f$ is constant. Can sb point me in the right direction?
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85 views
+50

conformal mapping, regions of the complex plane marked +/-, find the function f,

The picture shows what the function f: $\mathbb{C}\to\mathbb{C}\cup\infty$ does to the plane. The values 0 at 0, 1 at $\pm$1, and $\infty$ at $\pm i$ are specified. To elaborate on the picture: ...
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1answer
37 views

Analytic function on the whole plane, positive imaginary part, what can it be?

Part (a): The function f is analytic in the whole plane with positive imaginary part. What can it be? Part (b): What if all you know is that the imaginary part of f tends to 0 at infinity? what we ...
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0answers
40 views

Analytic Functions: Notation? [duplicate]

Analytic functions are usually denoted by $\mathcal{C}^\omega$. What does the $\omega$ stand for? (The infinity symbols of a colleague of mine really look like omegas...)
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1answer
27 views

A function property to guarantee that being constant on an interval implies identically constant

Let $f:\mathbb R\rightarrow \mathbb R$. Suppose we know that $f $ is a constant on some open/closed interval. Which condition does guarantee that $f $ is constant on $\mathbb R$? Clearly, continuity ...
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24 views

Solutions of complex equation for analytic function

Suppose $g(z)$ is an analytic function on the upper plane, in fact is constructed by Hilbert transform, \begin{equation} g(z) = g_R(x,y) + i g_I(x,y) \quad g_I = \mathcal{H}( g_R) + \text{real ...
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0answers
29 views

Complex analysis question, maximum principle application

Let $\Omega=\{z, \text{Re}z>0\}$ Suppose that $f$ is continuous in the closure of $\Omega$ and $f$ is holomoprhic on $\Omega$ and there are constants $A<\infty $ and $\alpha<1$ such that ...
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1answer
39 views

prove that a function is expressible as a power series

I started by rearrange f(z), and expanded the terms in summation. Then, I did not get very far. It would be great if anyone can let me know what is needed to figure out bn. Thanks in advance.
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1answer
48 views

Function holomorphic in the neighb. of zero, bounded by exponent is equal 0

I want to prove that if $f$ is a holomorphic function in a neighbourhood of $0$ and $|f(\frac{1}{n})| \le \frac{1}{e^n}$ for $n$ sufficiently big, then $f =0$. I know that if $f$ is holomorphic in a ...
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0answers
30 views

linear first order ODE with polynomial coeffficients

I'm wondering if anything can be said about the solution to a system of linear first order ODEs with polynomial coefficients, especially about analyticity of the solution. The equation is given as ...
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1answer
21 views

Local Estimates for higher order homogeneous elliptic operators

For $u\in W^{2k}_2(\mathbb{R^n})$, $k\geq 1$, it is well known (see, for example, Exercise 12.9.4 in Krylov, N. "Lectures on Elliptic and Parabolic Equations in Sobolev Spaces") that the following ...
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0answers
36 views

Preimage of the set of critical value of an analytic function between smooth manifolds

I have some problems with the following exercise, maybe due to alack of knowledge: Let $M$ be a connected smooth manifold and let $$ f \colon M \to N$$ be an analytic map. Denote by $C_f \subset M$ ...
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1answer
68 views

Suppose $f$ is entire and $|f(z)| \leq 1/|Re z|^2$ for all $z$. Show that $f $ is identically $0$.

This is a problem from my complex analysis textbook. The hint is to consider $g(z)=(z-iR)^2(z+iR)^2 f(z)$ and to show that $|g(z)| \leq 8R^2$. This is what i have tried: Consider $Re z \geq 0$, then ...
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0answers
21 views

Approximating an L2 function by analytic functions

Spose I have a $ h \in L^2(U) $ where $ U \subset \mathbb{R}^3 $ is open and bounded. Is it possible to approximate this by analytic functions? If so, spose now we take $ U = \mathbb{R}^3 $. Is this ...
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0answers
28 views

Real analytic function in one point and zeros

Let $f:I\to\mathbb{R}$ be a continuous function and $t_0\in I$ such that $f$ is analytic at $t_0$ and $f(t_0)=0$. Is it true that there is an entire neighbourhood of $t_0$ in which $f$ has no other ...
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1answer
10 views

Extend the domain of a real analytic function

Let $f:(a,b)\to \mathbb{R}$ be a real analytic function on $(a,b)$. Is there a real analytic function $g:(c,d)\to\mathbb{R}$, with $(c,d)\supset (a,b)$, such that: $ g(x)=f(x), \ \forall\ x\in (a,b)? ...
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1answer
39 views

If $f$ and $f\circ h$ are nonconstant and real-analytic, with $h\in C^\infty$, does it follow that $h$ is also real-analytic?

Let $f,g:I\to\mathbb{R}$ be two real, non-constant, analytic functions on an open interval $I\subset\mathbb{R}$. If there is a smooth function $h:I\to\mathbb{R}$ (i.e. $h\in C^{\infty}(I)$) such that: ...
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2answers
43 views

CR Equations using Polar Form

I have a question to check whether following function is analytic or not using CR Equations. The question is f(z) = 1/(z-z^5) I just don't know how to start and ...
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0answers
9 views

Analytic Continuation of Fourier Transform to a Strip in Complex Domain

This is to prove Theorem IX.13 from the Methods of Modern Mathematical Physics (by Reed & Simon). Let $f$ be in $L^2 (\mathbb{R}^n)$. Then $e^{b|x|} f \in L^2(\mathbb{R}^n)$ for all $b<a$, if ...
2
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1answer
41 views

Prove function is compex analytic

I want to prove that $f(z) = \frac{z}{e^z-1}$ is analytic around the origin. I tried using $z=x+iy$ and attempted to split $f$ into a $u(x,y) + iv(x,y)$ to apply the Cauchy Riemann equations, but this ...
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1answer
29 views

Can we find a real $u$ such that $f(u)=w$ ($w$ is fixed) and $f'(u)≠0$?

Let $f\colon\mathbb C\to\mathbb C$ be an entire non constant function. We consider its values on the real line. The function $f$ has infinitely many real zeros and there is infinitely many real ...
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30 views

Sufficient Conditions for Analyticity: C-R equations and?

My understanding of the subject is: (1) One of the necessary conditions for analyticity of f(x,y)=u(x,y)+iv(x,y) is satisfaction of the Cauchy-Riemann equations (in x-y, polar, or z-z* form). (2) The ...
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5answers
85 views

If $f=u+iv$ is an entire function such that $u^2\geq v^2,$ then $f$ is constant

Let $f=u+iv$ be an entire function such that $u^2(z) \geq v^2(z), \forall z \in \mathbb{C}.$ Could anyone advise me how to prove $f \equiv$ constant $?$ Hints will suffice. Thank you.
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1answer
86 views

Prove that f is one-to-one on D

Let $f$ be an analytic function on a disc $D$ whose center is the point $z_0$. Assume that $|f'(z)-f'(z_0 )|<|f'(z_0)| $ on D. Prove that $f$ is one-to-one on D.
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18 views

Complex Plane - Analytic Function

I am trying to understand the definition of an analytic function and how to solve for it's domain. I understand that for $f(z) = {1\over z}$ the function is analytic on the complex plane except for 0. ...
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1answer
26 views

Maximum Value - Analytic function

I am having a hard time figuring out where to start and what results to use to address the following question: Suppose $f(z)$ is analytic in the unit disc $D=\{z:|z|<1\}$ and continuous in the ...
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1answer
27 views

Minimum phase non-rational transfer function: Hilbert transform between log magnitude and phase

In Signal Processing literature, it is well known that a minimum phase sequence with rational transfer function ('zeros' and 'poles' in unit circle) has Hilbert transform relation between log ...
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1answer
17 views

A simple function with no tangential limits but with non-tangential limits

I am going to be teaching a course about the Hardy space and I would like to show the students that the non-tangential limit is a necessary concept BEFORE telling them about Blaschke products. Is ...
2
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3answers
71 views

Does the Taylor series of an infinitely differentiable function converge; and if yes, does it converge to the function? [duplicate]

I have googled it, but I am not satisfied with those. So my questions are: Let $D$ be an open set in $\mathbb{R}$. Let $f:D\rightarrow \mathbb{R}$ be a infinitely differentiable ...
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1answer
26 views

Is $\sum c_n z^n$ analytic when $c_n$ is Banach-valued?

I'm trying to define "Analytic function". I want a definition that covers all interesting cases. To be specific, let me explain what exactly I want Here is the definition of analytic function in ...
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1answer
17 views

Convergence of Taylor series about centre of open disc for analytic function.

I define a function on an open set of the complex plane to be analytic if about any point $z_0$ in that set it can be expanded as a power series in $(z - z_0)$ that converges in some neighbourhood of ...
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0answers
52 views

Analytic Continuation of the zeta function

Is the analytic continuation of the Riemann zeta function to the upper half plane unique? I don't know much complex analysis, so I can't see why that is the case.
2
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0answers
34 views

Proof that f(z) = $z^{n}(z^*)^{m}$ is not analytic in any point

Proof that f(z) = $z^{n}(z^*)^{m}$ is not analytic in any point. If i look at the limit of a more simple function of this form: f(z) = $\frac{z}{z^*}$ I would say that the limit does not exist, ...
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1answer
69 views

Can we characterize the space of functions which is real analytic but not real entire?

A complex valued function $F,$ defined on an open set $E$ in the plane $\mathbb R^{2}$, is said to be real-analytic in $E$ if to every point $(s_{0}, t_{0})$ in there corresponds an expansion with ...
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1answer
63 views

When is an analytic function in $L^2(\Bbb R)$?

Suppose $f:\Bbb R\to\Bbb C$ is real analytic. In order for $f$ to be in $L^2(\Bbb R)$, clearly all terms in the power series cannot be positive since $f$ would diverge at $\pm\infty$. Likewise, the ...
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57 views

Analytic approximations of the step function

Consider the Heaviside step function: $$H:\mathbb{R}\to \mathbb{R} $$ defined by $$H(x)=\begin{cases} 0 & \mbox{if } x<0 \\ 1 & \mbox{if } x\geq 0\end{cases}$$ Fix any $\delta>0$. Given ...
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1answer
30 views

Question about constructing complex entire function

How to construct an entire function for infinitely many prescribed values? i.e. I hope to find an entire function $f$ given $f(z_k) = w_k$ ($w_k$ might not be zero) for a sequence of $\{z_k\}$ with ...
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0answers
46 views

Determine a branch of $f(z)=\log(z^3-2)$ that is analytic at $z=0$ and find $f(0)$ and $f'(0)$

So I have found this problem in the last exam paper, but I have never done anything like this in class. The professor says that the resit will be very similar to the last 2 papers, so I would really ...
1
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1answer
22 views

Moment-determinacy in multivariate case

Let $X$ be a random vector with probability density $p$. In the scalar case I have learned that if the characteristic function of $X$ is real analytic, then all moments exist and $p$ is determined ...
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2answers
49 views

Complex function defined by contour integral along a smoothly varying path

Let $D$ be a domain in the complex plane. Consider the function $F: D\to \mathbb{C}$, defined by $$ F\left( z \right) = \int_{\mathscr{C}\left( z \right)} {f\left( {z,t} \right)dt} . $$ Suppose that ...
3
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1answer
44 views

Poles of Fourier transform

Let $f\in L_2(\mathbb R_+)$ and consider its Fourier transform $$F(\zeta)=\int_0^\infty f(x)e^{ix\zeta}dx$$ Is it true that analytic continuation of $F(\zeta)$ has at most finitely many poles in a ...
5
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1answer
102 views

A real analytic function that takes each value in $\mathbb{R}$ three times

I was inspired by this question: it is quite easy to prove that for any positive odd number $2m+1$ there exists a function $f\in C^{\infty}(\mathbb{R})$ such that ...
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1answer
54 views

Upper bound for modulus of a function

Let $f(t,x)$ be a bounded and continuous function on $\mathbb{R}_t \times \mathcal{U}$ where $\mathcal{U}$ is an open neighborhood of $0 \in \mathbb{C}_x$. Moreover, assume that for each fixed $t$, ...
0
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1answer
35 views

Does analytic at a point implies differentiable at that point?

A function $f:A \subset \Bbb C \to \Bbb C$ is said to be analytic if it is locally given by a convergent power series i.e. every $z_o \in A$ has a neighbourhood contained in $A$ such that there exists ...
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0answers
50 views

Zeros of the derivatives of a finite Blaschke product.

Let $B$ be an $n$ degree finite Blaschke product. By considering the level curves of $B$, one can show that $B'$ has $n-1$ critical points in the disk (counting multiplicity). Is anything known ...