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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48 views

An Elliptic function can not be holomorphic/analytic?

I was reading about elliptic functions on the wiki and it said that a doubly periodic meromorphic function in contention of being an elliptic function can not be analytic/holomorphic as it would then ...
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37 views

Can a meromorphic function which does not have any poles in a domain be shown to be bounded in that domain?

Basically such a function would not have any singularities in that domain and is completely analytic on it. Then, is the function bounded on that domain?
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68 views

Example of an analytic continuation for a function in integral form

Given $f(z) = \int_{-\infty}^\infty \frac{exp(-t^2)}{z-t}\,dt$, where $Im(z)>0$. Find an analytic continuation to the region $Im(z)<0$. Firstly the solution said that there is a branch cut on ...
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1answer
43 views

Does analytic continuation apply only to analytic functions?

I'm a high school senior attempting to do a project on the riemann zeta function. I've looked online, tried reading college textbooks but still don't have a completely clear idea of analytic ...
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56 views

Is the determinant an analytic function?

I came accross a paper stating that the analytical property of determinants of complex matrices allows us to use some theorem for analytic functions. I am not able to confirm this since I am not sure ...
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0answers
18 views

Is definite integral of such function multiplication analytic?

If $f(x)$ is a general function (integrable) and $g(s,x)$ is an analytic function except for on its poles. Then, can some one judge about $$H(s)=\int_{a}^b f(x) g(s,x) dx $$ Is $H(s)$ analytic ...
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1answer
40 views

The radius of convergence of a power series about a point interior to the domain of an analytic function

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a real analytic function with domain an open, non-empty set $(a, b) \subseteq \mathbb{R}$, $-\infty \leq a < b \leq \infty$ and let $c \in (a, b)$. ...
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1answer
63 views

Prove that $f$ analytic, $f(x) \in \mathbb{R}$ for all $x \in \mathbb{R}$ implies $f(\overline{z})=\overline{f(z)}$

Let $U\subset \mathbb{C}$ be a nonempty connected open set such that for every $z\in U$, $\overline z\in U$. Let $f$ be analytic on $U$. Suppose $f(x)\in\mathbb R$ for every $x\in U\cap\mathbb R$. ...
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1answer
96 views

If $f$ is analytic, prove that $\overline{f(\overline{z})}$ is also analytic

Let $f$ be an analytic function in an open set $U \subseteq \mathbb{C}$. Let $V=\{z\in\mathbb C:\overline z\in U\}$. Define $g$ on $V$ by $g(z)=\overline{f(\overline{z})}$. Show that $g$ is analytic ...
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49 views

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? ...
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51 views

Analytical solution to nonlinear ode

I solved this equation that I attached numerically in matlab by the Newton Raphson method. Now I want to solve it analytically in matlab or even in Maple if it is possible. Would you please help me ...
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1answer
104 views

Computing Complex Integral to Determine Analytic Continuation of $f(z) = \int_0^\infty {{e^{-zt}} \over {1 + t^2}} dt$

My question is the following: Find the analytic continuation of the function $f(z)$ defined by $$ f(z) = \int_0^\infty {{e^{-zt}} \over {1 + t^2}} dt, \ \ \vert \arg(z) \vert < {1 \over ...
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0answers
50 views

Why is $1/z$ analytic at infinity?

I was given this proof: Let $w(z)=1/z$, so $w$ maps origin to inifinity and infinity to origin. Consider $f(z) = z$. It has no singularities in finite $z$-plane. So $f(w) = 1/w$ has a pole at the ...
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30 views

Analytic branch of root

I'm trying to prove that if $f$ is holomorphic function in an open subset $G$ of the complex plane, and $z_0 \in G$ is a zero of $f$ of order $m$ then there exist a branch of the $m$th root of $f$ - ...
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1answer
25 views

Finding all the analyitical function in the unit annulus that satisfy a given condition for natural numbers

Let $f$ be an analytic function in the annulus $0 < |z| < 1 $ such that it's singularity in $z=0$ is not essential. I want to find all of such functions $f$ that satisfy for $n = 3, 4,...$: ...
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21 views

Showing that $n^z$ for $n\in\mathbb{Z}$ and $z\in\mathbb{C}$ is analytic

I just had a quick question. Can I just say that because we know $n^m$ for any integer $m$ is entire, and since we know that $z$ is entire, then the composition, $n^z$ is entire?
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2answers
79 views

A uniformly convergent sequence of real analytic functions which does not converge to a real analytic function

I am looking for an example of a uniformly convergent sequence of real analytic functions which does not converge to a real analytic function. Also I would appreciate any pointers on how to think of ...
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1answer
88 views

If all the roots of a polynomial P(z) have negative real parts, prove that all the roots of P'(z) also have negative real parts

If all the roots of a polynomial $P(z)$ have negative real parts, prove that all the roots of the derivative $P'(z)$ also have negative real parts. Could anyone provide a proof for this please?
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0answers
72 views

Real analytic function with radius of convergence 1 at non-negative integers

So, as the title states, the problem I was confronted with was to find a real-valued everywhere analytic function $$f:\mathbb{R}\to \mathbb{R}$$ s.t. at every non-negative integer, k ...
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1answer
33 views

Holomorphic function is zero on an analytic set then $df=0$.

Assume we have an homomorphic function $f:U\rightarrow \mathbb{C} $ which is holomorphic on the open set $U$ of $\mathbb{C}^n$. Assume there is $V\subset U$ analytic and that $f$ restricted to $V$ ...
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29 views

What are conditions for an infinite sum with a complex parameter not to be analyitically extendable?

I'm looking for a sequence $f(n)$, so that $g(z):=\lim_{N\to\infty}\sum_{n=0}^N\exp\left(-z\cdot f(n)\right),$ with $z$ so that this converges classically, defines a function which can not be ...
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1answer
30 views

Determining if a function is real anaytic at the point $a$?

Is there a method, other then using refer to the Taylor series to determine if a real function is analytic at the point $a$. If so please, if possible, could you give a source.
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35 views

Single value function defined in the plane cut

I'm confused about multi/single value complex function. Can someone explain why the function $\sqrt{1-z^2}$ can be thought of as a single valued in the plane cut along $-1\leq z\leq 1$
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1answer
46 views

Existence of such function

So we know that if $g(z)=\frac{z-c}{1-\overline{c}z}$ $(c\in\mathbb{C})$ $|g(z)|=1$ for $|z|=1$. Does there exist a function $f(z)$ satisfies the following properties: (1) $f$ is analytic in some ...
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1answer
18 views

Sequence of partial sums converges locally uniformly?

Suppose $f: U \to \mathbb{R}$ is a real analytic function defined by $f(x)=\sum_1^\infty a_n (x-x_0)^n$ and let $f_N=\sum_1^N a_n(x-x_0)^n$. Then $\{f_N\}$ converges to $f$ pointwise. Wikipedia says ...
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0answers
107 views

Proof of the Riesz-Schauder Theorem (for compact operators) using the Analytical Fredholm Theorem

First of all sorry for my bad English, I'm an Italian student, hope to let you understand! I'm having a little troubles with the proof of the Riesz-Schauder theorem for Compact Operators. Some infos ...
2
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1answer
100 views

An entire and one-to-one function must be of the form AZ+B, A non-zero. How to rule out higher degree polynomials in z? [duplicate]

Show that if f is entire and one-to-one, then it must be of the form AZ+B, with A not equal to zero. I am editing my question, since there are duplicates on this forum to the question of why an ...
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2answers
99 views

Is any rational function $R(x)$ a real analytic function in its domain?

To begin with, the definition of a rational function $R(x)$ can be found in Wiki. Suppose that $R(x)$ is defined in a subset $D \subseteq \mathbb{R}^n$. Then my question is: Is any rational ...
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46 views

Existence and uniqueness of a function generalizing a finite sum of powers of logarithms

I hope to find a proof of the following conjecture: $(1)$ For every $a>0$ there is a convex analytic function $f_a:\mathbb R^+\to\mathbb R$ such that: $f(1)=0$ and $\forall x>1,\ ...
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1answer
88 views

Proving analytic continuation, choosing suitable branch cuts,

Consider the function $$f(z)=\log[(z^2+1)^{1/2}],\quad z>0$$ where the branch is chosen so that $(z^2+1)^{1/2}>0$ for $z>0$ and the log denotes the principal branch. Let $R$ be the union of ...
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1answer
63 views

how to prove that a function is not complex differentiable

I was working on a problem on the complex differentiability of the following function: $f(z)= z \operatorname{Re}(z)$. How to find the points where the given function is not differentiable. My ...
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33 views

Prove that risk function is analytic?

I'm considering the statistical minimax estimation problem of the bounded normal mean: Specifically, the problem is to find the minimax estimator of $X \sim N(\theta,1)$ where $\theta \in ...
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0answers
39 views

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, $$ ...
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2answers
54 views

Real valued and holomorphic function

I was wondering about this problem for a while out of curiosity: is there a non-constant analytic function with real values on $\mathbb{R}$ and purely imaginary values $i\mathbb{R}$? I think the ...
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0answers
59 views

Number of connected components of $f^{-1} (U)$

Let $f:\mathbb{R}^n \to \mathbb{R}$ be an analytical function (semialgebraic,polynomial if needed), $U$ be an open connected subset of $\mathbb{R}$. What can we say about the nuber of connected ...
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112 views

Why are numeric methods the only technique available to solving $\ln(x) = \sin(x)$? Is this $x$ transcendental?

I just read this question about finding the solution to the equation $\ln(x) = \sin(x)$. All the answers focus on using a numerical method to approximate the solution. This is interesting in its own ...
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1answer
91 views

Are the Cauchy-Riemann equations a necessary and sufficient condition for a function to be analytic?

If we have a region R is $f(z)$ analytic in the region R if and only if it satisfy the Cauchy-Riemann equations for every point in R. If not what are the other conditions it must satisfy? Do we have ...
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2answers
88 views

Proving Polynomial is Analytic

If a function $f$ at $x = a$ equals it's Taylor Series, $f$ is said to be analytic. So, if I were given a polynomial $p(x) = \sum_{n=0}^{200}{a_nx^n}$, and trying to prove that $p(x)$ was analytic ...
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0answers
32 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 ...
5
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1answer
127 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
82 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 ...
4
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2answers
137 views

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

Which entire holomorphic functions satisfy $\,\lvert\,f(z)\rvert \leq \lvert z\rvert^k$, for all $z\in\mathbb{C}$? So I've shown that $\,\lvert\, f(z)\rvert \leq \lvert z\rvert ^k \implies f(z)$ is a ...
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2answers
106 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?
3
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1answer
229 views

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
79 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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1answer
36 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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0answers
53 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 ...
1
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0answers
53 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 ...
2
votes
1answer
50 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.