Questions tagged [analytic-functions]

For questions about analytic functions, which are real or complex functions locally given by a convergent power series.

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38 views

Functional equation $F(x+1) = x + F(x)$ and analytical expansion of Riemann zeta-function

I have found a solution for functional equation $F(x+1) = x + F(x) \tag1$ $F(x) = \frac{x(x-1)}{2}+C$, where $C$ - some constant However, I have also tried to recursively expand equation $(1)$ at $0$...
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30 views

Analytic function on the complex plane except the negative real axis

Let $G$ and $H$ be defined by $G=\mathbb{C}\setminus\{z=x+iy: x\le 0, y=0\}$ $H=\mathbb{C}\setminus\{z=x+iy: x\in \mathbb{Z},x\le 0, y=0\}$ Suppose $f:G\to \mathbb{C} $ and $g:H \to \mathbb{C}$ ...
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1answer
32 views

Determine analytic function outside the unit circle from value of real part at the boundary.

I'm addressing the following problem. Let $G(z)$ be an analytic function outside of the unit circle with decay at infinity and Re $G(z)$ at $|z|=1$ some known Laurent polynomial (with poles only at $...
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21 views

A Schwarz lemma like exercise, inequality [duplicate]

Let $D_r=\{z\in \mathbb{C} | |z|<r\}$ be the disc of radius $r$, for $r>0$. Let $f$ be an analytic function on $D_2$. Then (i)Prove that there exists a constant $C>0$ such that $|f^{'}(z)|\...
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22 views

Complex function with values on the unit circle copied everywhere

If $f:\mathbb{C}\setminus \{0\}\to \mathbb{C}$ is a function such that $f(z)=f(\frac z{|z|})$ and its restriction to unit circle is continous,then $(1)\lim _{z\to 0} f(z)$ exist. $(2)f$ is analytic ...
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1answer
37 views

An application of Rouché's theorem

i need help to proof the next: Let $f$ be analytic at $D$ minus a finite numbers of interior points where $f$ has poles. Show that if $0<|f(z)|<1$ over $\partial D$, then the number of poles of ...
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30 views

Bounding an analytic function on the closed unit disk [duplicate]

I have just encountered the following exercise which has me quite stumped: Let $ f $ be analytic on the closed unit disk, and we assume that $ | f(z) | \leq 1 $ for all $ z $'s in this set. We also ...
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32 views

Cauchy product between a power series and a vector of power series

From here, it is known that given any formal power series in the following form ($a_0$ may or may not be zero), $$g(z) = \sum_{i=0}^\infty a_i z^{i} \quad \quad (1)$$ there exists another non-...
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1answer
26 views

Prove that all singularity of $\frac{1}{e^z+3z}$ is of order 1

This is a problem from my past QUal: "Prove that all singularity of $$\frac{1}{e^z+3z}$$ is of order 1. You don't need to find the singularities." Usually this kind of problem is easy to me. My ...
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28 views

A subsequence of $f^n$ converge pointwise where $f$ is analytic on the unit ball

This is a problem from my past Qual: "Let $\Omega$ be the unit disk and $f:\Omega\to \Omega$ be an analytic function. COnsider the sequence $\{f^n(z)\}$ where $f^n=f\circ\ldots\circ f$ ($n$--times). ...
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1answer
52 views

$f:D\to D$ is analytic then $f^{n_i}(z)$ converges pointwise for all $z$

This is a problem from my past Qual. "Let $D$ denote the unit disk and $f:D\to D$ be analytic. Show that there exists a sequence $n_i$ s.t. $f^{n_i}(z)$ converges pointwise for all $z\in D$. Here $f^...
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1answer
36 views

Cauchy product of two formal power series

I am thinking if I could get help for the following question: Given a formal power series $$g(z)=\sum_{i=0}^\infty a_i z^{-i}$$ does there always exists another (non-trivial) formal power series $...
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24 views

pole and zero cancellation for power series

I typically work on rational functions. There, the pole and zero cancellation can be done straightforwardly. For example, consider $$ G(z)= \frac{z-1}{z+2}, $$ I can cancel the pole $-2$ of $G(z)$ by ...
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1answer
26 views

Help with exercise in complex analysis on the existence of a mapping

I have encountered the following exercise from a practice exam: Does there exist an analytic function mapping the annulus: $ A = \{ z | 1 \leq |z| \leq 4 \} $ onto the annulus: $ B = ...
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2answers
46 views

A sequence $a_1=f'(0),a_2=f''(0),…$

I am working on this problem from my past Qual "Give a sequence s.t. there is no analytic function $f:D\to \mathbb{C}$ s.t. $a_1=f'(0),a_2=f''(0),...$" where $D$ is the unit disk." The only thing I ...
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1answer
25 views

the family of analytic functions with positive real part is normal. [duplicate]

I'm reviewing Complex Analysis and I don't quite understand the concept of normal family. There is an exercise in Ahlfors' Complex Analysis: Prove that in any region the family of analytic functions ...
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26 views

$f(z^n)=f(0)+g(z)^n$ in a neighborhood of $0$

I'm working on this problem If $f(z)$ is analytic in $|z|<1$ and $f′(0)\neq 0$ prove that there exists an analytic function $g(z)$ such$f(z^n)=f(0)+g(z)^n$ in a neighborhood of $0$. There is a ...
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40 views

A holomorhic function $f $ in the unit disc such that $\lim_{z\rightarrow 1}f(z)$ does not exist

Let $f$ be a holomorphic function in the open unit disc such that $\lim_{z\rightarrow 1}f(z)$ does not exist. Let $\sum_{n=0}^{\infty}a_nz^n$ be the Taylor sereis expansion of $f$ about $z=0$ and $R$ ...
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40 views

Complex analysis $f(z)=(1+z^2)^i$ [closed]

If $f(z)=(1+z^2)^i$ basic function branch, draw at the complex plane the subset that is analytic.
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42 views

Are there references about universal power series and this statement?

The notion of universal power series is linked with this statement : We can find a power series $\sum_{n\ge1} a_nx^n$ such that for all continuous function $f : [0,1] \to \mathbb{R}$ such that $f(0)...
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17 views

Non-constant analytic function on an open connected subset $U$ of $\mathbb{C}$

Let $U\subset \mathbb{C} $ be an open connected set and $f:U\rightarrow \mathbb{C}$ be a non-constant analytic function. Consider the following sets:- $X=\{z\in U:f(z)=0\}$ $Y=\{z\in U:f $ ...
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1answer
28 views

Analytic function on an open subset $U$ of $\mathbb{C}$

Let $U$ be an open subset of $\mathbb{C}$ and $f:U \rightarrow \mathbb{C}$ be an analytic function.Then which of the following are true? $(a)$ If $f$ is one-one, then $f(U)$ is open in $\mathbb{C}$ ...
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1answer
32 views

derivative of hypergeometric function

I am doing an integral in Mathematica and I find the solution contains derivatives of hypergeometric functions. I would like (ideally) a simple analytic form for these. I have tried HypExp mathematica ...
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1answer
11 views

Regular singular, Irregular singular & Ordinary points of an equation

I'm currently trying to re pick up physics after a few years off.. but I'm not sure on the current question and I have no idea what it actually means. My mathematics is not great but it reminds me of ...
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1answer
38 views

zeros of a function in a simple closed contour

i would really appreciate if you can help me with the following demonstration please: Let $C$ be a closed simple contour such that $| f (z) | = cte$ ($f$ is analytic inside $C$ and is not an ...
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1answer
15 views

Derivative of function f at origin provided 2Ref + 3Imf = 1.

Let $f = u + iv$ be analytic in a connected open set. If $u$ and $v$ satisfies $2u + 3v = 1$, then what will be the value of $f'(0)$ ? Since $f$ is analytic we know it satisfies Cauchy Riemann ...
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1answer
23 views

Use Schwarz' Lemma and/or Maximum Modulus Principle to Prove This Proposition

Suppose the following conditions hold true: (1) The function $f$ is analytic and contractive in the open unit disk. (2) $f(0)=0$. (3) $\exists z_1 \ne z_2 \in B(0,1)$, $|z_1|=|z_2|$, $f(z_1)=f(z_2)$...
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73 views

Dilogarithm of a negative real number outside unit circle

The dilogarithm is defined in $\mathbb{C}$ as $$ Li_2(z) = -\int_0^1 \frac{\ln(1 - zt)}{t} dt $$ Because $1-zt \in \mathbb{C}$, then you can write $\ln(1 - zt) = \ln|1 - zt| + i·\arg(1 - zt)$ As ...
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0answers
29 views

Analytic function's proof verification - I didnt use all the given condition. Where's the mistake?

let $ f\left(x\right)=\sum_{n=0}^{\infty}a_{n}x^{n} $ be with positive convergence radius $ R>0 $ and assume that exists sequence $ y_n \in(0,R) $ decreasing monotonic such that $ y_{n}\to0 $ and ...
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1answer
64 views

Holomorphic function$f: D \to D$ with $f(\frac{1}{2}) =-\frac{1}{2}$ and $f '(\frac{1}{4}) =1$

Does there exist a holomorphic function$f: D \to D$ with $f(\frac{1}{2}) =-\frac{1}{2}$ and $f '(\frac{1}{4}) =1$ where $ D= \{ z \in \mathbb{C} : |z|<1\}$. I cannot use any of Schwarz lemma or ...
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0answers
24 views

Given a smooth, non analytic curve, obtain a non analytic smooth function.

I managed to prove (by methods of complex analysis) that the curve $$f:[0,1]\to \mathbb{C};\\ x\mapsto e^{2\pi ix}+\sum_{n=5}^\infty \frac{\left(e^{2\pi ix}\right)^{(2^n)}}{n!}$$ is a smooth, non ...
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1answer
61 views

Analytic function $\sum\limits_{n=1}^{\infty}\frac {z^{2n-1}}{2n-1}.$

Let $f:\{z:\|z\|<1\}\rightarrow \{z:-\frac{\pi}4<\operatorname{Im}(z)<\frac {\pi}4\}$ such that $f (z)=\sum\limits_{n=1}^\infty\dfrac{z^{2n-1}}{2n-1}$. How to prove that $f(z)$ is analytic ...
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2answers
41 views

How to prove that $ \frac{1}{x^{k}} $ is analytic in $ \mathbb{R} \setminus [0] $

I want to prove that $ \frac{1}{x^{k}} $ is analytic in $ \mathbb{R} \setminus [0] $ for |x|<1 its obvious. i want to prove for the rest of $ \mathbb{R} $ so i tried that: I proved by ...
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1answer
16 views

Complete the last step in an analytic function proof

let $ f$ be analytic function in segment A. and assume $ \in A $ and $ f(0)=0 $ prove that exists $ n \in N $ and analytic function $ g(x) $ in segment A such that: $ f(x)=x^n\cdot g(x) $ for ...
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1answer
25 views

A trivial question about power series (analytic functions)

let $ f\left(x\right)=\sum_{n=0}^{\infty}a_{n}x^{n} $ absolute convergent series with convergent radius $ R>0 $ Assume $ a_0,a_1,a_2,...a_{m-1}=0 $ so $ f\left(x\right)=\sum_{n=m}^{\infty}a_{n}x^...
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1answer
33 views

proving the function $ \frac{1}{\left(x-2\right)^{2}} $ is analytic

So, I want to prove that the function $ \frac{1}{\left(x-2\right)^{2}} $ is analytic in $ \mathbb{R} \setminus [2] $ What I've tried: let $ I\in\mathbb{R}\setminus\left\{ 2\right\} $ be open sigment ...
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0answers
27 views

Use of the Identity Theorem?

I have the following question: Let $s(y)$ and $t(y)$ be real differentiable functions on $y$ with $-\infty < y < \infty$ satisfying $s(0) = 1$ and $t(0) = 0$, with the property that the ...
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1answer
33 views

Formula of an ellipse-like curve with custom skew and min/max points

I need to figure out the formula for a function of one variable x that looks like an ellipse, but I have control on points: min $x = A$ max $x = B$ the point $x'$ that curve attains its maximum value ...
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0answers
24 views

Why is this function defined by iteration analytic?

The function I'm dealing with is reached by solving iteratively the following integral equation ($x$ is real, and $ u $ is a real smooth function that goes to zero on infinity): $$ \chi(x) = 1-\int_{...
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2answers
80 views

Does the ring of analytic functions have zero divisors?

Question I have to show that the ring of complex analytic functions on open unit disk has no zero divisors. My attempt let suppose $fg≡0$ such that $f≢0$ and $g≢0$ on open unit disk $U$ then $f$ and ...
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3answers
395 views

Why does $f(z) = z^n$ have no antiderivative only for $n=-1$?

The complex valued function $f(z) = z^n$ has an analytic antiderivative on $\mathbb{C} \setminus \{0 \}$ for every $n$ except for $n=-1$. What is so special about $-1$? To show why this is such an ...
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2answers
95 views

If $f$ increasing, analytic on $\mathbb{R}$ and $\lim_{x\to +\infty}f(x)=1$, does it follows that $\lim_{x\to +\infty}f'(x)=0$?

Question: If $f$ strictly increasing, analytic on $\mathbb{R}$ and $\lim_{x\to +\infty}f(x)=1$, does it follows that $\lim_{x\to +\infty}f'(x)=0$? If we drop the assumption that the function is ...
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1answer
48 views

Pointwise convergence of holomorphic functions on a dense set

Let $G$ be an open connected set and let $D \subset G$ be a dense set. Let $(f_n)$ be a sequence of holomorphic functions in $G$ and assume $f_n \rightarrow 0$ pointwisely on $D$. Can we deduce that $...
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2answers
27 views

Finding analytic function with given condition

I have this task from complex analysis: Find analytic function $ f(z) $ such that $ |f(z)|=e^{{\rho}^2\cos(2\theta)} $ where $ z=\rho e^{i\theta}. $ I'm guessing I should use Cauchy-Riemann conditions,...
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2answers
45 views

Check the continuity and analyticity of complex function $f(z)$

Given function $$\displaystyle f(z)=\left\{\begin{matrix} \frac{\bar{ z}^2}{z} & \text{if} \;\; z \neq 0\\ 0 & \text{otherwise} \end{matrix}\right.$$ I have to check its continuity and ...
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2answers
29 views

Find all values of a complex constant c where a function is analytic in $\mathbb{C}$

Let $f:\mathbb{C} \to \mathbb{C}$ be a function defined by $f(x+iy) = x^2-y^2+2cxy$ , where $c$ is a complex constant. In other words, $c=c_1+ic_2 \in \mathbb{C}$. We also define $x,y,c_1,c_2 \in \...
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0answers
50 views

Showing $\zeta(s)-{1\over s-1}$ is analytic

It is well known that Euler-Mascheroni constant has an alternative definition in terms of zeta function: $$ \gamma=\lim_{s\to1^+}f(s)\equiv\lim_{s\to1^+}\left[\zeta(s)-{1\over s-1}\right] $$ Using ...
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3answers
56 views

Analytic gradient $\nabla f$ implies analytic $f$

I'm wondering how to prove that a real function $f : \mathbb{R}^d \to \mathbb{R}$ with analytic gradient $\nabla f$ (equivalently, analytic Fréchet derivative $Df$) must also be analytic. We can ...
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1answer
39 views

If $f(z)$ is analytic in a domain D then show that $f^2(z)$ is analytic there.

If $f(z)$ is analytic in a domain D then show that $f^2(z)$ is analytic there. Let $f(z)=u+iv$, then $f^2=u^2-v^2+2iuv=X+iY$, say then by CR equation due to analyticitiy, it can be shown that CR ...
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0answers
28 views

Meaning of “analytic function” in the context of Banach algebras

In going through Folland's Abstract Harmonic Analysis, I came on the following. Let $\mathcal{A}$ be a unital Banach algebra with unit $e$, and define $$\sigma(x) = \{ \lambda \in \mathbb{C} : \lambda ...

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