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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2answers
33 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
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3answers
91 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
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2answers
49 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
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2answers
27 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
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0answers
62 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
45 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
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0answers
31 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
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2answers
56 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 ...
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1answer
37 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
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1answer
63 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
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0answers
10 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
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1answer
44 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
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1answer
30 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
46 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
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0answers
32 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
37 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
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2answers
64 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 ...
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2answers
32 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
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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
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0answers
36 views

I found a harmonic function from a convergent Laurent series; is this harmonic function unique?

I am guessing that it is simply, "yes", since the Laurent coefficients are unique. I solved for the coefficients to get the Laurent series, showed that it converges, and then took the real part of ...
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0answers
31 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
38 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
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0answers
36 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
26 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 $2 k \pi i, \ \ k \in \mathbb{Z}$. ...
4
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2answers
119 views
+100

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
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1answer
33 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 ...
2
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1answer
28 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 ...
2
votes
1answer
31 views

Complex inequalities and constants

Let $f(z)$ be an analytic function. Show that if $|f(z)| > 1 + |e^z|$ then $f(z)$ is constant. I have no idea what to do, I've subbed $z= x+iy$ and got $|f(z)|>1+e^x$ but lost here.
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2answers
25 views

Finding analyticity

Given that f is analytic, under what conditions is $g(z)=\overline{f(z)}$ analytic? Does this explanation make sense? : $g'(z)=lim_{h\rightarrow 0} \dfrac{g(z+h)-g(z)}{h}=lim_{h\rightarrow ...
0
votes
1answer
24 views

If $F$ is analytic and injective on the unit disc and $B(F(0), |F'(0)|)\subseteq F(B(0,1))$, then $F(z)= F(0) + F'(0)z$.

Let $F$ be analytic and injective on $B(0,1)$. Show that if $B(F(0), |F'(0)|)\subset F(B(0,1))$, then $F(z)= F(0) + F'(0)z$. I have tried the following: Since $F$ is injective we know that ...
1
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0answers
25 views

Multisection of a Power Series Proof

Suppose that $$H_{N,k}(x)=\frac{x^ke^{\frac{-x}{N}}}{N^{k-1}k!\sum_{n=0}^{N-1}{w_N^{-nk}e^{\frac{w_N^nx}{N}}}}=\sum_{n=0}^\infty{A_n\frac{x^n}{n!}}$$ where $k\lt N, w_N=e^{\frac{2i\pi}{N}}$, and ...
0
votes
3answers
33 views

Show that there exists $g: \mathbb{\Omega} \rightarrow \mathbb{C}$ analytic such that $g(z)^n = f(z)$ for all $z \in \mathbb{\Omega}$

I'm learning about complex analysis and need some help with this problem: Let $\mathbb{\Omega}$ open, simply connected, $f: \mathbb{\Omega} \rightarrow \mathbb{C}$ analytic without zeros in ...
0
votes
1answer
33 views

Prove a function to be analytic by dominated convergence theorem

Given $f \in L^1$, prove that $$F(z) = \frac{1}{2\pi i} \int^\infty_{-\infty} \frac{f(t)}{t-z}\,dt$$ is an analytic function and $$F'(z)=\frac{1}{2\pi i} \int^\infty_{-\infty} ...
2
votes
1answer
26 views

Cesaro limit of analytic functions

Let $f_n$ be a uniformly bounded sequence of analytic functions on $\Omega\subset\mathbb C$. If $f_n(z)\to f(z)$ forall $z\in\Omega$, then by the Montel's theorem I know that the convergence is ...
0
votes
1answer
47 views

Is my idea of decomposing a meromorphic function into a sum of Laurent series correct?

We know that complex-analytic functions $f(z)$ agree with their power series representations on their domain of analyticity. If a meromorphic function has simple poles at $z_1, ..., z_m$ and ...
2
votes
1answer
25 views

If $f$ is analytic and $f(0)=a_0$ then prove that $|\lambda| \ge \frac{|a_0|}{M}$.

Let , $f$ be non-constant holomorphic function in a nbd. of $\bar{\mathbb D}$ with $f(0)=a_0$. Let , $\displaystyle M=\max_{z\in \mathbb D}|f(z)$. Let, $\lambda \in \mathbb D$ and $f(\lambda)=0$. ...
1
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2answers
40 views

How can I show that this meromorphic function is a rational function of two polynomials?

Here's my updated attempt: Write$$f(z) = \sum_{n=-1}^{\infty} a_n(z-z_1)^n + ...+\sum_{n=-1}^{\infty} m_n(z-z_m)^n+\sum_{n=+1}^{-\infty} \psi_n(z)^n$$ with the last series being an expansion about ...
1
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0answers
31 views

Is this result for entire functions with zeroes only at the origin… more basic than the Hadamard canonical product representation?

I just worked on a problem and was able to solve it pretty easily, using Hadamard's product representation. But I wonder whether the solution that I compared my work to doesn't actually use the ...
0
votes
0answers
10 views

Bounded analytic functon with small derivatives

A question from the theory of bounded analytic functions. Let $f$ be analytic in the circle $D: |z|<1$ and bounded in $D$ by absolute value by a constant $M>0$. We assume that $N$ derivatives ...
1
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1answer
32 views

Prove that if $\overline {D_1}\subset f(\Omega ) $ then $\overline {D_r} \subset f(\Omega )$ for some $r>1$

Let $f:\Omega \to \mathbb C$ be an analytic function such that $\Omega $ is an open set.Define $D_r=\{z:|z|<r\}$ for some $r>0$. Prove that if $\overline {D_1}\subset f(\Omega ) $ then ...
1
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1answer
20 views

Bounded holomorphic function, limsup, boundary

Let $f \in \mathcal{O}(D)$, $\ D$ is a bounded region in $\mathbb{C}$, be such that $$\limsup _{D \ni z \to z_0} |f(z)| < \infty$$ for any $z_0 \in \partial D$. Prove that $f$ is bounded. So we ...
2
votes
0answers
72 views

How can I conclude that f(z) is constant? [duplicate]

I'm struggling a bit to arrive at the conclusion that $f(z)$ is a constant. Suppose $f(z)$ is holomorphic on and inside the unit disk, and that it has no zeroes on the interior. Also, assume that ...
0
votes
1answer
15 views

Is this a continuity / connectedness argument or is it an orientation-preservation argument?

Take, for example, the simple linear fractional transformation that sends the upper half plane to the unit disk, and the real line to the unit circle. We know the fact that the upper half plane (UHP) ...
0
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1answer
26 views

When computing contour integration with sines and cosines in the integrand, must we always first look at Euler's formula?

For example, in computing $$\int_{Cr}\frac {\cos(z)}{(z^2+a^2)^2}dz$$ over a semi-circular contour, must I first look at $$\int_{Cr}\frac {e^{iz}}{(z^2+a^2)^2}dz$$ compute this integral first, ...
1
vote
1answer
31 views

How can I compute the residue at this order-2 pole?

The integral is $$\int_{-\infty}^{\infty} \frac {cos(z)}{(x^2+a^2)^2}dz $$ If I use an upper semi-circular contour, then there is an order-2 pole at $z=ia$. I am trying to expand the integrand in a ...
1
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0answers
16 views

Can an analytic function defined on a maximal torus be extended analytically to all the Lie group?

Let $G$ be a compact group and $T$ a maximal torus on $G$. Suppose $f$ is an analytic function defined on $T$. Is there an analytic function $F$ on $G$ whose restriction agrees with $f$ on $T$?
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0answers
18 views

Regarding the Schwarz Reflection Principle, is getting the analyticity of f(z) on the real axis a consequence of the theorem itself,

or a consequence of Morera's Theorem? Basically, I want to be able to cite it correctly, e.g., can I say we have not only continuity of f(z) along $R$ (by assumption) but also it turns out that ...
1
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1answer
39 views

Distance preserving transformations of the complex plane

Show that the most general transformation fixing the origin and preserving distances is either a rotation, or a rotation followed by a reflection in the real axis, for a transformation $f: \mathbb{C} ...
1
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0answers
33 views

Show that an analytic function defined on unit ball with these properties does not exist

Show that there does not exist an analytic function $F : B(0,1) \to \mathbb{C}-\mathbb{Q} $ with $F(0)=i$ and $F(1/2)=-i$. I have already used the open mapping theorem to show that if we assume ...
1
vote
1answer
24 views

Show that if $f(z)$ is a continuous function on a domain $D$ such that $f(z)^N$ is analytic for some integer $N$, then $f(z)$ is analytic on $D$.

Show that if $f(z)$ is a continuous function on a domain $D$ such that $f(z)^N$ is analytic for some integer $N$, then $f(z)$ is analytic on $D$. I had no clue, I was trying to use the facts that ...