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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2
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1answer
34 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

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
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1answer
42 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 ? $1.\large\sum_\limits{n=0}^{\infty}\...
0
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0answers
43 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
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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
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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
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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 |...
2
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1answer
34 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
31 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}\dfrac{\...
0
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1answer
38 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 $F'(0)\...
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0answers
30 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
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3answers
50 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 $\...
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1answer
45 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} \frac{f(t)}{(t-z)^2}\...
2
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1answer
28 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
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1answer
65 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
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1answer
28 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 $f(\lambda)=0$. ...
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vote
2answers
59 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 ...
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0answers
40 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 ...
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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 ...
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1answer
38 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
35 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
76 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 $|...
1
vote
1answer
24 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
29 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, ...
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1answer
34 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 ...
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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
21 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 $f(z)$...
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1answer
44 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} \...
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0answers
42 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 ...
2
votes
1answer
51 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 ...
6
votes
2answers
155 views

Determine the complex function $f\left ( z \right )$

The details provided are that the function is analytic and that its real part along the line $y=c x$ is constant. What conclusions can I draw from here? I think that this imposes $\frac{\partial u}{\...
0
votes
1answer
65 views

Why real valued harmonic functions are holomorphic.

Let $f$ be a real valued harmonic function on $C,$ then Claim $g= \frac {\partial f}{\partial x} - i \frac {\partial f}{\partial y} $ as holomorphic and $h= \frac {\partial f}{\partial x} + i \frac {\...
0
votes
1answer
49 views

Why is a harmonic conjugate unique up to adding a constant?

If $v$ and $v_0$ are harmonic conjugates of $u$, then $u + iv$ and $u + iv_0$ are analytic functions. Then $i(v - v_0)$ is analytic, but how does this imply $v - v_0$ is a constant function?
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2answers
90 views

Example of real analytic function

We were taught real analytic functions in class today. I am playing around trying to construct examples. I see exponential, sine, cosine and logarithmic functions (for $x > 0$). One function I am ...
0
votes
1answer
34 views

Proove $|f(z)|\leq M|z|$ inside unit disk if $f(0)=0$ and $|f(z)|\leq M$

The problem goes as follows: Assume $f(z)$ is analytic inside the unit disk $D=\{z:|z|<1\}$. Also, $f(0)=0$ and $|f(z)|\leq M$ in $D$. Proove that $|f(z)|\leq M|z|$ in $D$. When does ...
0
votes
1answer
50 views

Proving that a complex function is analytic, and finding its power series

Let $I \subseteq \mathbb{R}$ be an interval and $g: I \to \mathbb{C}$ continuous. Define $f: \mathbb{C} \backslash \overline{Im(f)} \to \mathbb{C}$ by $f(z) := \int_I \frac{1}{g(x) - z} dx$ (with $\...
0
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1answer
43 views

Zeros of real analytic functions on $\mathbb{R}^n$

Consider a non-constant multivariate real analytic function $f$ on $\mathbb{R}^n$. My question is, can the zeros of $f$ be dense in $\mathbb{R}^n$? In one dimension, I know that they cannot be, as the ...
0
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1answer
35 views

Prove convergence of $\int^{\infty}_{0}\ t^{z-1}cos(t)dt$ and $\int^{\infty}_{0}\ t^{z-1}sin(t)dt$

For a complex analysis problem set I am trying to show that the integrals $$\int^{\infty}_{0}\ t^{z-1}cos(t)dt \quad and \quad \int^{\infty}_{0}\ t^{z-1}sin(t)dt $$ is convergent for $0<Re(z)<...
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0answers
89 views

$f(x+1)=f(x)+f(\alpha\cdot x)$

I try to find an analytic increasing function $f_\alpha$ ($0\le\alpha\le1$) from $\mathbb R$ (or $\mathbb R^+$) to $\mathbb R$ such that for all $x$ $$f_\alpha(x+1)=f_\alpha(x)+f_\alpha(\alpha\cdot x)$...
2
votes
3answers
51 views

Approximation of non-analytic function

I have a function which is of the form \begin{equation} f(x) = \frac{1 - x^{1/2} + x - x^{3/2} + \ldots}{1+x^{1/2} - x + x^{3/2} - \ldots}. \end{equation} Intuitively, I would assume that for small $x$...
0
votes
1answer
39 views

Proove that $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y}$ is the real part of an analytic function.

The problem is as follows: Proove that $U(x,y) = x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y}$ is the real part of an analytic function. where $f(z)$ is analytic such that $...
3
votes
1answer
23 views

Is sudden appearance of poles possible?

I was thinking about the following scenario: For $|\epsilon|<1$ let $f_\epsilon(z)$ be a meromorphic function such that $f_0(z)$ has a pole at the origin for $\epsilon \neq 0$, $f_\epsilon(z)$ ...
0
votes
2answers
69 views

Does there exist a holomorphic function with the following property?

Does there exist a holomorphic function $f$ defined over $D = \{ z : |z| < 1 \}$ such that $|f| \rightarrow \infty$ when $|z| \rightarrow 1$? My approach: If such an $f$ exists, then for a given ...
1
vote
1answer
25 views

analytic deformation of a compact set in the complex plane

Let $K$ be an uncountable compact set in $\mathbb{C}$ such that zero is a limit point of $\partial K$, and such that $|k|\leq 1$ for all $k\in K$. I would like to find an analytic function $f:U\to\...
0
votes
0answers
51 views

Mittag-Leffler theorem for real analytic functions

I just started reading about the Mittag-Leffler theorem which says that given an open set $G \subset \mathbb{C}$, and a sequence of distinct points $a_k$ in $G$ (without a limit point), if we denote $...
3
votes
1answer
38 views

Does Cauchy's estimate imply analyticity?

Komatsu says here (Proc. Japan Acad. Volume 36, Number 3 (1960), 90-93) that a smooth function which satisfies Cauchy's estimate is analytic. How does one prove this? Surely, if Cauchy's estimates ...
0
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0answers
24 views

Reasoning in extra poles from analytic continuation

Entertain the following situation: Let $w(z)$ be a complex function which is a single valued function of $z$ inside the unit circle $|z|<1$. The derivative $\frac{dw}{dz}$ has simple poles at the ...
2
votes
1answer
52 views

Is continuity of first partials required for analyticity?

Let's cast the complex function $f(z) = u(z) + iv(z), z = x+iy$, as the multivariable function $F(x,y) = U(x,y) + iV(x,y) ; x,y \in R$. Thus, $$dF = F_x\,dx + F_y\,dy = U_x\,dx + iV_x\,dx + U_y dy + ...
0
votes
2answers
40 views

Is my explanation correct regarding Maximum value of Sine function over $\Bbb C$?

Question: What is the maximum value of sine function taking domain as $\Bbb C$? My answer is: The maximum value is not defined. Explanation: Since the range of sine function is $\Bbb C$ and $\Bbb C$ ...