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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4
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1answer
53 views

Approximating polynomials over $\mathbb{C}$ with an entire function

Given a series of polynomials $p_{n}$ and a series of non-intersecting balls $B_{n} \subset \mathbb{C}$ show that there exists some function $f \in \mathcal{O}(\mathbb{C})$ such that $lim_{n ...
3
votes
1answer
100 views

Analytic function on unit disk has finitely many zeros

I am studying complex analysis from Theodore Gamelin's text and Exercise 1 of chapter IX.2 says that if $f$ is analytic inside the open unit disk and continuous on its boundary that satisfies $|f(z)| ...
1
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2answers
46 views

Some proofs about the infinite products; the series and the analyticity

Definition A.1.4. The infinite product $\prod_{n=1}^{\infty}(1+a_{n}(x))$, where $x$ is a real or complex variable in a domain, is uniformly convergent if $p_{n}(x)=\prod_{m=k}^{n}(1+a_{n}(x))$ ...
3
votes
1answer
39 views

Analyticity of solutions to the heat equation

Let us look at solutions to the linear heat equation on $\mathbb{R}$: $$ u_t = u_{xx}.$$ Is it true that solutions to the equation with nice enough initial datum are analytic after a certain time $T ...
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0answers
33 views

Is the solution of a polynomial an analytic function on the polynomial parameters?

Be $\mu(z_1, \ldots, z_L)$ the only positive real solution to the equation \begin{equation} \sum_{l=1}^L z_l \mu^l = 1 \end{equation} With $z_1 = 1$, $z_l \geq 0 \forall l$. Clearly, varying the ...
2
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1answer
23 views

Prove: Show $\frac{|f'(z)|}{1-|f(z)|^2}<\frac{1}{1-|z|^2}$ for $f(z)=z^n$ and any $n \in N$

Prove that if $f:\mathbb{D} \to \mathbb{D}$ (where $\mathbb{D}$ is the unit disk) is given by $f(z)=z^2$, the for all $z \in \mathbb{D}$, we have $$\frac{|f'(z)|}{1-|f(z)|^2}<\frac{1}{1-|z|^2}$$ ...
0
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0answers
28 views

Holomorphical extension to the Annulus

Let $D=\{z:1<|z|<2\}$ and $f$ is holomorphic on $D$. Suppose that f has a primitive $f_1$ on D and $f_1$ also has a primitive $f_2$, etc for every $n$ $f_n$ has a primitive $f_{n+1}$ in $D$. How ...
2
votes
1answer
16 views

Suppose $\{f_n\}$ is a sequence of analytic in a region $D$

suppose $\{f_n\}$ is a sequence of analytic in a region $D$ such that $|f_n(z)|\leq M_n$, where $\sum_n M_n<\infty$, and $\lim_{z\to z_0}f_n(z)=L_n$. Show that if $p=p(z)\to\infty$ as $z\to z_0$, ...
1
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0answers
48 views

Suppose $g$ is a continuous function such that the integral $\int_0^\infty|g(t)|dt$ converges

Suppose $g$ is a continuous function such that the integral $\int_0^\infty|g(t)|dt$ converges. Is $$F(z)=\int_0^\infty g(t)\sin(zt)dt$$ analytic? And if so, in what region? My attempt: ...
1
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0answers
79 views

Simply connected domains and complex logarithms

While studying Complex Analysis from my professor's notes I came across the following theorem. A demain $D$ in the complex plane is simply connected if and only if any analytic function $f(z)$ on ...
1
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2answers
57 views

If $f^{\prime}(z)=g^{\prime}(z)$, then $f(z)=g(z)+c$

Suppose $f$, $g:G \to \mathbb{C}$ are defined on a domain $G \subseteq \mathbb{C}$ and are differentiable on $G$. Then, if $f^{\prime} = g^{\prime}$, then $f = g + C$ for some constant $C$. I ...
0
votes
1answer
35 views

$f(1/n)=1/n$ implies $f(z)=z$

If $f:\mathbb{C}\to\mathbb{C}$ is entire and satisfies $f(1/n)=1/n$ then $f(z)=z$. I am trying to prove this. What I have so far: by continuity $f(0)=0$. Since $f(1)=1$, it feels tempting to show ...
0
votes
2answers
76 views

If the real part of $f$ is bounded then $f$ is constant

It isn't too hard to show that if $f:\mathbb{C}\to\mathbb{C}$ holomorphic everywhere (entire) and $\Re (f)$ is bounded, then $f$ is constant: it suffices to consider $\exp f$, which is entire, and by ...
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0answers
19 views

Composition of analytic functions is analytic in Manifolds

My problem is in analytic manifolds.According to Cohn's book a function $f$ in a manifold $M$ is analytic at $p \in M$ if it can be expressed as a power series of $\sigma(p)=(x_{0})$. That means ...
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0answers
21 views

$D:=\{z \in \mathbb C: |z|<1\}$ and $f:D \to \mathbb C$ be an analytic function such that $|f(z)|\le 1-|z| , \forall z \in D$ , then $f=0$? [duplicate]

Let $D:=\{z \in \mathbb C: |z|<1\}$ and $f:D \to \mathbb C$ be an analytic function such that $|f(z)|\le 1-|z| , \forall z \in D$ , then is $f$ identically zero ?
2
votes
1answer
27 views

If $f,g$ real analytic and $\lim_{t \to t_0} f(t)/g(t)$ exists then $f/g$ is analytic

If $f,g$ are real analytic at $t_0$ and $\lim_{t \to t_0} f(t)/g(t)$ exists then is it true that $f/g$ with the limiting value filled in at $t= t_0$ is real analytic at $t_0$? I know the complex ...
13
votes
0answers
168 views

Is this function nowhere analytic?

One usually sees $f(x):=\exp\frac{-1}{x^2}$ as an example of a $C^\infty$ function that is not analytic, having one point of non-analyticity (the point $0$). The Fabius function is a canonical ...
0
votes
1answer
38 views

Finding analytic function

I am trying to solve the following problem, "Show that if $h(z)$ is a complex-valued harmonic function such that $zh(z)$ is also harmonic, then $h(z)$ is analytic." My approach was to calculate first ...
1
vote
1answer
27 views

Clearer definition of a singularity?

"A point $z$ is said to be a singularity of the function $F(z)$ if in the complex plane there exists no circle with center at $z$ within which $F(z)$ is analytic." Can someone describe this a little ...
0
votes
2answers
64 views

Taylor series expansion of $ f(x)=e^{-x^2}$

How to find Taylor series expansion of $f(x)=e^{-x^2}$ What I'm stuck at is proving that the error $$R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}$$ of the expansion tends to zero.
4
votes
2answers
100 views

Which meromorphic functions are logarithmic derivatives of other meromorphic functions?

Let $f$ be a meromorphic function defined on the whole complex plane. Is there a characterization in terms of easier-to-test properties of $f$ whether or not $f=g'/g$ for some entire $g$? The ...
3
votes
0answers
54 views

Show that $f(z) = \ln r + i \varphi$ is differentiable in a neighborhood of $z_{0}$

I am faced with the following problem: Let $z_{0}\neq 0$ and let $f(z) = \ln r + i \varphi$, where $r = |z|$, $\varphi \in arg z$, and $\varphi$ is chosen so that $f$ is continuous in a neighborhood ...
0
votes
0answers
15 views

Analytic structures which induce same topology on $\mathbb{R}$

It an exercise in Cohn's book. Analytic structure on a Hausdorff space $M$ is a family of charts $\mathcal{F}$ satisfying At each point of $M$ there is a chart in $\mathcal{F}$ Any two charts of ...
1
vote
1answer
49 views

Show that $g$ is analytic and discuss the properties of $g$

Let $f$ be analytic in $\overline{B}(0; R)$ with $f(0)=0$, $f'(0) \neq 0$ and $f(z) \neq 0$ for $0<|z| \leq R$. Put $\rho=\min\{|f(z)|:|z|=R\}>0$. Define $g: B(0; \rho) \rightarrow \mathbb{C}$ ...
1
vote
2answers
60 views

Analytic function $f,$ such that $f(0) = 1$ and $f'(z) = zf(z),$ for all $z \in \mathbb{C}$

I'm trying to find an example of an analytic function $f$ satisfying the IVP $$ f'(z) = z\,f(z), \quad f(0) = 1, $$ and for all $z \in \mathbb{C}$, but I'm somewhat at a loss of the best way to ...
4
votes
3answers
128 views

If $f: \mathbb{C} \rightarrow \mathbb{C}$ is analytic and $\lim_{z \to \infty} f(z) = \infty$ show that $f$ is a polynomial

I'm learning about complex analysis and need some help with this problem: If $f: \mathbb{C} \rightarrow \mathbb{C}$ is analytic and $\lim_{z \to \infty} f(z) = \infty$ show that $f$ is a ...
0
votes
2answers
58 views

Does there exist an analytic function $f$ on $D(0,1)$ such that $f(z_n)=0$ for even $n$ and $f(z_n)=1$ for odd $n$?

Given that $(z_n)$ is a sequence of distinct points in $D(0,1)=\{z \in \Bbb C : |z| \lt 1\}$ with $\lim_{n \to \infty} z_n=0$, Can we find an analytic function $f$ such that $f(z_n)= \begin{cases} 0, ...
0
votes
2answers
36 views

A function that satisfies $|f(z)-\bar z|<0.9$ is not analytic in the unit circle.

I've came accros this excersize: Suppose that $D=\{z:|z| \le 1\}\subset \mathbb C$ and $$f:D\rightarrow\mathbb C$$ suppose that for every $z\in D$ such that $|z|<1$ $$|f(z)-\bar z|<0.9$$ where ...
5
votes
0answers
69 views

Finding an explicit entire function $g$ satisfying $g(n \log n) = n^{\pi}$

I encountered the following problem in the lecture note in my complex analysis class: Problem. Find an explicit entire function $g$ satisfying $g(n \log n) = n^{\pi}$ for $n = 1, 2, \cdots$. ...
3
votes
1answer
55 views

Is there an upper bound on the growth rate of analytic functions?

This problem comes from a solution tactic used in Is there a rational surjection $\Bbb N\to\Bbb Q$?, where I discovered that there is an analytic function $f(z)$ that takes the values $f(n)=a_n$ for ...
2
votes
0answers
34 views

Verification of example to show surjective maps of sheaves need not surject onto sections in all open sets

As an exercise in understanding the notion of surjectivity in the category of sheaves, I came up with this example, slightly modifying the standard ones given in my textbooks. I feel like this one is ...
-4
votes
2answers
57 views

$e^{\mathrm{Re}\,z}$ not analytic in complex plane

In my textbook I found a text where it says that $e^z$ (z is a complex number) is analytic everywhere. But $e^x=e^{\mathrm{Re}\,z}$ is not. How can I prove that about $e^x$ and what is the ...
-1
votes
1answer
40 views

Example of two analytic functions that differ at countably infinity many point

$\displaystyle f_1(x) = \frac{x^n-1}{x-1}$ and $f_2(x) = x^{n-1} + \cdots + 1$ have the same values except at $x=1$ (where $f_1$ fails to be analytic ). Is there an example of two analytic function ...
0
votes
1answer
106 views

If $f(z)$ is meromorphic but not entire, is $\exp(f(z))$ meromorphic? Could it even be entire?

First, I can show that $f$ meromorphic is a rational function. Now, I want to consider $g=e^{f(z)}$. I have heard that there is something interesting that goes on with $g$, that there is some room ...
0
votes
0answers
12 views

Is the rank of a matrix-valued mapping $\mathbb R^n \rightarrow \mathbb R^{n_1 \times n_2}$ constant almost everywhere?

How can I prove the following statement or even better: which source can be quoted for the proof. Consider an $n_1 \times n_2$ matrix $A(z)$ where $z\in \mathbb{R}^n$ is arbitrary and the the entries ...
1
vote
1answer
49 views

Clarification on Riemann Mapping Theorem

A comp exam problem asks us to prove "the following piece of the Riemann Mapping Theorem." If $f,g$ are analytic bijections from an open set $A$ to the unit disc $D$ with $f(a) = g(a) = 0$ and ...
1
vote
1answer
31 views

$f$ analytic on $\mathbb{C}-\left\{0\right\},$ $\text{Im}f<-1$ for $|z| = 1/2,2$; show $f(1)\neq 0$.

Suppose $f$ is analytic on $\mathbb{C}-\left\{0\right\}$ and that $ \Im(f) < -1$ for $|z|=1/2$ and $|z| = 2$. Show $f(1)\neq 0.$ I have tried Cauchy's Integral formula, but it seemed to be a dead ...
1
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2answers
51 views

Greatest common divisor of real analytic functions

Consider two real-valued real analytic functions $f$ and $g$. I want to prove that there exists a greatest common divisor $d$, which is a real analytic function. By greatest common divisor, I mean the ...
3
votes
0answers
55 views

Branch points and Riemann surfaces (analytic continuation),

Take probably the most typical example: $$f(z) = \sqrt{1-z^2}$$ This function uses the (complex) logarithm to define it: $$e^{\large \frac{1}{2}log(1-z^2)}$$ $$e^{\large \frac{1}{2}[ln|1-z^2| + ...
2
votes
1answer
39 views

Largest neighborhood on which $f(z)=\sum\limits_{p\text{ prime}} z^p$ defines an analytic function

Find the largest $r$ such that $$f(z) = \sum\limits_{p\text{ prime}}z^p$$ defines an analytic function on $B_r(0).$ The series diverges for $|z|\geq 1$, since the terms don't go to zero, and ...
3
votes
2answers
71 views

Constructing a function with N zeros inside the unit disk,

Find all functions $f(z)$ such that: a) $f$ is analytic in some region containing |z| $\le$ 1 b) $|f| = 1$ on $|z| =1$ c) $f$ has N simple zeros $z_1, ... , z_N$ inside $|z| < 1$ and no other ...
1
vote
2answers
34 views

Can the complex square root of $z\sin z$ be defined in a neighborhood of the origin? (I.e., including the origin)

Edit: on a second thought, I don't think it's possible since $$ f(z) = \sqrt {z\sin z} = e^{\large \frac{1}{2} \log z}e^{\large \frac{1}{2} \log\sin z}$$ $$e^{\large \frac{1}{2} (\ln|z| + ...
2
votes
1answer
33 views

Recommendation of a good source on Lyapunov theorem in dynamical systems

As part of my research I wish to read a full proof of Lyapunov's classic theorem on dynamical systems that for an analytic planar vector field where all Lyapunov/focal values are zero, the local phase ...
0
votes
0answers
35 views

Convergence and analyticity of functions

My question is, generally, is there any relationship between convergence and analyticity of a complex-valued function (namely, does one property imply the other etc?) For example, a function defined ...
0
votes
1answer
39 views

Solving the following inequalities in entire function [duplicate]

Let $f(z) = \large\sum_\limits{n=0}^{\infty}\normalsize a_n z^n\:$ be an entire function and let $\:r\in\mathbb{R}$. Which of following inequalities hold ? ...
0
votes
0answers
42 views

Laurent series $\frac{1}{e^z-1}$ [duplicate]

How can I expand $$f(z)=\frac{1}{e^z-1}$$ into Laurent series? I know that $f$ has singularities in $2k \pi i, \ \ k \in \mathbb{Z}$. Just substituting Taylor series for $e^z$ in the denominator ...
0
votes
1answer
27 views

Singularities of difference of two functions

We are given $$f(z) = \frac{1}{\sin z} - \frac{1}{e^z-1}$$ $\frac{1}{\sin z}$ has poles in $k \pi, \ \ k \in \mathbb{Z}$ and $\frac{1}{e^z-1}$ has singularities in $2 k \pi i, \ \ k \in \mathbb{Z}$. ...
3
votes
1answer
194 views

An entire function bounded outside a strip which contains the reals is constant

Let $f$ be an entire function, which takes real values on the real axis and has no zeros. Suppose $f$ is bounded for $|\operatorname{Im} z| > a > 0$ where $a>0$. Is $f$ a constant? I would ...
0
votes
1answer
36 views

Analytic functions and constants: Proving $f(z)-g(z)$ is constant

Let $f(z)$ and $g(z)$ be analytic on some domain. Show that if $\Re(f(z)) = \Re(g(z))$ then $f(z)-g(z)$ is constant. I haven't a clue on how to start. What is being asked of me & What am I ...
3
votes
1answer
39 views

Mapping the region $\Gamma_{z}$ using the conformal map $ \omega=\frac{-2z}{z^{2}+1}$

Suppose we have an analytic function $$ \omega=\frac{-2z}{z^{2}+1}$$ and the region $\Gamma_{z}$ given by $$\Gamma_{z}:=\left \{ z \in \mathbb{C}| \Im \left ( z \right )\geq 0 \wedge \left | z \right ...