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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Real analyticity of function of two variable

I have a question concerning the following claim: let $\rho,C,r>0$, and consider $$ v(x,y) = (\rho-x) - \sqrt{(\rho-x)^2- 2Cr y}, $$ then the function $v$ is real analytic in the domain $D=\{(x,y) \...
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Question about about the proof of Jensen Formula (Conway)

Let $F(z) = f(z)\prod_{k=1}^{n}\frac{r^{2}-\overline{a_{k}}z}{r(z-a_{k})}$ which is analytic on $\overline{D}(0;r)$ and has no zero in $D(0;r)$. I don't understand why $\vert F(z)\vert =\vert f(z)\...
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Proof of Bloch's theorem: continuity of $h(r)$

Bloch's Theorem. Let $f$ be an analytic function on a region containing the closure of the disk $D= \lbrace z ; |z|<1 \rbrace$ and satisfying $f(0)=0$ and $f'(0)=1$, then there is a disk $S\subset ...
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Find Taylor series for $\cosh z \cos z$

Find Taylor series for $\cosh z \cos z$. $\cos z = \cosh iz$ and $\cosh z \cosh iz = \dfrac{1}{2}(\cosh (i+1)z + \cosh (i-1)z)$ and finally $$\cosh z \cos z = \dfrac{1}{2}\sum_{n=0}^\infty {\dfrac{(i+...
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Description of complete analytic function for $\sqrt{4z-\sqrt[3]{z}}$.

The problem is to describe all branches and all the curves of analytic continuation of $\sqrt{4z-\sqrt[3]{z}}$. I started with representing function in a way $\sqrt{4w^3-w}\circ\sqrt[3]{z}$ So, there ...
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When do real, analytic, monotonic functions on an interval extend to univalent functions on an open region of the complex plane?

Suppose I have a real, analytic function $f(x)$ which is monotonic on some connected, open subset of the real line $W \subseteq \mathbb{R}$ and such that $f'(x)>0$ on $W$. Let me naturally include ...
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Estimate on derivative of ODE solution with respect to parameters

Consider the ODE $$ u'(t) = f(t,u,p), \qquad u(0) = v $$ where $p$ is a control parameter, and let $u(t;v,p)$ denote the solution to the problem above for fixed $v$ and $p$. It is apparently "...
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Find the limit of the integral of analytic function [closed]

To solve this problem, I used the triangle inequality and hints 1 and 2. But when I used hint $2$, I got the limit of the absolute value of the given integral, which is not zero. so I can't find the ...
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Number of zeros of a power series

Consider the infinite series $$f(z) = 2\sum_{k=0}^\infty(-1)^k z^{(2k+1)^2} -1.$$ I want to show that $f$ admits one zero in the interval $(0,1)$ (in $\mathbb{R}$). I have perfect knowledge about the ...
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Books/online notes for analytic maps in Banach spaces

Can anyone suggest some good books/online notes on (real) analytic maps between Banach spaces? I am looking for the basic definitions and the implicit function theorem in this setting. Thanks!
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Problem using Liouville's Theorem [closed]

True/ False There does not exists function $f$ that is analytic in $D$ such that $|f(z)| \leq 1, f\left(\frac{1}{3}\right)=0$ and $f\left(\frac{-2}{3}\right)=\frac{5}{6}$ If $D=\mathbb C$ then by ...
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Analytic tanh-like function with infinite radius of convergence.

Consider the Taylor expansion of $\tanh$ around $0$. The radius of convergence is finite ($\pi/2$). Define a $\tanh$-like function a function $f:\mathbb R\to\mathbb R$ such that: $f(0) = 0$; $\lim_{x\...
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Can a function be analytic only inside a given curve??

A function can be analytic only inside a circle. For example the function $g(z)= 1+z+z^2+...$ is only analytic inside the unit circle and nowhere else. This example can be modified to fit into any ...
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If there exists $M>0$ such that $|f(z)|>M$ for all $z \in \Bbb Z$, show that $f$ is constant in the whole complex plane. [duplicate]

Let $f$ be an analytic function in the whole complex plane. If there exists $M>0$ such that $|f(z)|>M$ for all $z \in \Bbb Z$, show that $f$ is constant in the whole complex plane. I tried to ...
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Proof of the possibility of introducing isothermal parameters

I’m currently reading differential geometry in the large by H. Hopf. On page 99 Hopf writes “a simple proof of the possibility of introducing isothermal parameters can be given providing E, F, G are ...
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Give an example which shows that a function $f$ need not be complex differentiable at $z_0$.

If we have a analytic map $f: D(0,1) \to \Bbb C$ on the whole disc $D(0,1)$, then prove that the Cauchy-Riemann equations hold on all points of $D(0,1)$. Furthermore suppose that the Cauchy-Riemann ...
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Continuing $\log(z)$ analytically along the curve $\gamma(t)=e^{it}$

Show that $\log(z)=\sum_{n=1}^\infty \frac{(-1)^{n-1}(z-1)^{n}}{n}$ can be analytically continued along the curve $\gamma(t)=e^{it}$. Compute $f_t$ and their radii of convergence. What is $f_{2\pi}$? ...
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Do analytic function and power series agree whenever the power series converges?

Let $f:\mathbb{R}\to\mathbb{R}$ be a function which is analytic at $x=0$. Then $$f(x)=\sum_{n=0}^\infty a_nx^n$$ for all $x$ in a neighborhood of $0$. Let $R$ be the radius of convergence of $\sum_{n=...
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Finding Coefficients of an Analytic Function

For what values of $a, b, c$ can $u(x,y)=ax^2+bxy+cy^2$ be the real part of an analytic function? Obviously $0$ works, but that's not the interesting case. I think we're to make use of the the CR ...
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A subgroup $G$ of the group of analytic isomorphisms of the open unit disc $Aut(D(0,1))$ is the entire group

Suppose $G$ is a subgroup of $\text{Aut}(D(0,1))$ which contains all the origin -fixing rotations and at least one element other than these. Then the problem asks me to prove that $G$ is all of $\text{...
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Conditions for $\Gamma_f(\lambda) = \int P^{\lambda}f$ to be analytic in $\Re (\lambda) >0$

In his book about Rings of Differential Operators, Björk mentions the following: Let $f\in C_0^{\infty}(\mathbb{R}^n)$, and let $P(x_1,...,x_n)$ be a real valued and $\geq 0$ polynomial. We define the ...
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'Proof' checking of anti-derivative of an analytical function

My 'Proof': If $f\in H(\Omega)$ ,then $f(z)=\sum\limits_{n=0}^{\infty} c_n(z-a)^n,\forall z\in D(a;R)\subset \Omega$ . Pick $F(z)=\sum\limits_{n=1}^{\infty}\frac{c_{n-1}}{n}(z-a)^n(\forall z\in \Omega)...
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Characterizing functions which satisfy Cauchy integral formula

Charecterize all complex valued functions $f$ satisfying below statement: Suppose $C$ is a simple closed curve inside a simply connected domain $D$ and for any point $a$ inside $C$, $$2i\pi f(a)=\...
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1 answer
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How do I compute the following integral using Cauchy's integral formula?

Let $f$ be an analytic function on a neighborhood $\{|z|\leq 1\}$ and $\gamma$ is a unit circle oriented counterclockwise. Show that for $0<|z|<1$ $$2\pi i f(z)=\int_\gamma \frac{f(w)}{w-z}~dw-\...
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Find all analytic function on $\mathbb{D}$ satisfying a certain condition

Find all analytic functions on $\mathbb{D}$ s.t $f(\frac{1}{n})+f''(\frac{1}{n})=0$. My attempt: Using uniqueness theorem, I'm looking for all analytic functions satisfying $f=-f''$. This implies ...
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Composition of analytic function with non analytic function.

I have the following question. If I have two complex valued functions, $f,g$. I know that if both are analytic then $f\circ g$ is analytic (if domain/codomain matches). Now I thought about if $g$ is ...
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Compactly supported Fourier transform for "patching function"

Let $g:\mathbb{R}\backslash\{0\}\to\mathbb{R}$ be the function $g(x)=\frac{1}{x^2}$ and let $I=[-c,c]$, for some $c>0$. I would like to know if there exists a function $\varphi:\mathbb{R}\to\mathbb{...
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Does the analytic continuation method for infinite series always work and is unique?

Sorry if this is worded poorly, but I don’t know enough in the subject to word it better: $$\sum_{n=0}^{\infty}n=-1/12$$ according to $$\zeta(-1)=\sum_{n=1}^{\infty}\frac{1}{n^s}=-1/12$$ Is there ...
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If $f(x)$ is analytic and non-negative, does convergence of $\int_a^{\infty} f(x)\,dx$ imply $\lim_{x\to\infty} f(x)=0$?

It is well known that improper integrals don't have to satisfy $\lim_{x\to\infty} f(x)=0$ in order for $\int_a^{\infty} f(x)\,dx$ to converge, for instance $f(x)=\sin(x^2)$. It is also possible to ...
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How do I check that this function is analytic using the derivatives?

I have the following problem. We define $\Omega=\Bbb{C}\setminus \{x:x\in (-\infty 0]\}$. For $z=x+iy\in \Omega$ we define the curve $\gamma_z$ going first from $1$ to $1+iy$ and then from $1+iy$ to $...
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Isn't analyticity needed for the schwarz reflexion principle?

I asked myself if analyticity on the upper half-line is needed in the analytic version of the Schwarz reflexion principle. Or can I only say that if $f:\{Im(z)\geq 0\}\rightarrow \Bbb{C}$ is ...
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3 votes
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How do I show that the following function is analytic using the Morera theorem?

Let $f:\Bbb{C}\rightarrow \Bbb{C}$ be a continuous function which is analytic on $\Bbb{C}\setminus \Bbb{R}$. I need to use the Morera theorem to show that $f$ is analytic on $\Bbb{C}$. In the lecture ...
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Given $u(x,y) = e^x (x \cos y - y \sin y)$ for analytic $f(z) = u + iv$, find $v(x,y)$

Given $u(x,y) = e^x (x \cos y - y \sin y)$ for analytic $f(z) = u + iv$, find $v(x,y)$. I know that there is an answer to this question in For an analytic function $f(z)=u+iv$, if $u=e^x(x \cos y-y \...
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3 answers
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to prove a Holomorphic function to be constant

Suppose that $f$ is holomorphic on $\mathbb{C}$, and suppose that the function $g(z) = f(z)/z$, defined for $z \neq 0$, satisfies $g(z) → 0$ as $|z| \to \infty$. Prove that $f$ is constant. I have ...
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2 votes
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Number of zeros of an entire function.

Let $f$ be an entire function then my question is: a formula to find the number of zeros of $f$ on the line $\Re(s)=1,0<\Im(s)<h$ where $h>0$. I was thinking of applying the argument ...
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Why can I pull out the maximum of an integral?

I have the following problem. We have $R\subset \Bbb{C}$ a rectangle and $\{f_n\}$ a collection of analytic functions which converges uniformly to $f$. After a long computation we got to the ...
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Question about Bloch's Theorem : Exercise XII.1.2 from Functions of one complexe variable (John B. Conway)

Bloch's Theorem. Let $f$ be an analytic function on a region containing $\overline{\mathbb{D}}$ and satisfying $f(0)=0$, $f'(0)=1$. Then there is a disk $S\subset \mathbb{D}$ on wich $f$ is one-one ...
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How do I explain how the analytic function looks like?

I have the following problem. We have given $g$ to be an analytic function such that for $|z-z_0|<R$ we have $$|g(z)|\leq M$$. We assume that $g$ have a zero of order $m$ at $z_0$. Then I have ...
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Question about Lemma1.1 to prove Bloch's Theorem

1.1 Lemma Le $f$ be an analytic in $D=D(0,1)$ and suppose that $f(0)=0$, $f'(0)=1$, and $\vert f(z)\vert \leq M$ for all $z$ in $D$. Then $M\geq 1$ and $f(D)\supset B(0, 1/6M)$. Proof. Let $0<r<...
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Proof of Bloch's theorem: property of $h(r)$

Bloch's Theorem. Let $f$ be an analytic function on a region containing the closure of the disk $D= \lbrace z ; |z|<1 \rbrace$ and satisfying $f(0)=0$ and $f'(0)=1$, then there is a disk $S\subset ...
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2 answers
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Do holomorphic functions have primitive?

Let $f:\Omega\to \mathbb C$ be called holomorphic on $\Omega$ if it is complex-differentiable on $\Omega$.Define primitive of $F$ on $\Omega$ to be a function such that $F'(z)=f(z)$ for all $z\in \...
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Prove or disprove: analytical function

Prove or disprove: There exist an analytical function $f(z)$ in a deleted neighbourhood of $z=0$ such that $f^2(z) = z$. I disproved it with the following argument: the function $f(z)$ is given by ...
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How can I show this fact about analytic functions?

I have the following problem: Let $f,g$ be analytic functions when $|z|<2$. Let us define $D=max\{|f(z)|+|g(z)|: |z|\leq 1\}$. Let us suppose that $M=|f(w)|+|g(w)|$ for some $w$ with $|w|\leq 1$. ...
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How do I prove that there is a complex logarithm?

I have the following problem: Prove that there exists an analytic function $$f:\{z\in \Bbb{C}:|z-1|<1\}\rightarrow \Bbb{C}$$ such that $e^{f(z)}=z$ when $|z-1|<1$. We have just shown this fact ...
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  • 1,031
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Compute the path integral $\int_{0}^{i}{\sin(z)}dz$

I'm pretty stumped on this question. This is what I've got so far. $\int_{0}^{i}{\sin(z)}dz = -{\cos(i)} + {\cos(0)} = 1 - {\cos(i)}$. I am using the following theorem: Let $D\subseteq \mathbb C$ be ...
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How can I show that an analytic function can be bounded from below as follows?

I have the following problem: Let $f$ be a nonzero analytic function on $B:=\{|z|<1\}$. Show that there exists $c>0$, $r>0$ and $k=0,1,2,...$ such that $$|f(z)|\geq c|z|^k$$when $|z|<r$. ...
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2 votes
2 answers
103 views

Solving $k\left(e^{a(2k+1)}-1\right)=1$ for k?

I've been studying the structure of partial sums of the Dirichlet eta function and noticed that certain critical points occur when the summing from $n=1$ to $n\approx N(k)$ for $k\in\mathbb{N}$ where $...
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complex function continuous but not differentiable on an open set

Could one possibly come up with an example of a complex-valued function such that $f(z)$ is continuous on the open set $D=\{ z \in \mathbb{C}:z \neq 0 \}$ and $\lim\limits_{z\to 0} f(z) = \infty$ but ...
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Proving that $\lambda$ is not analytic in differential topology.

Let $\lambda : \mathbb{R} \to \mathbb{R}$ be defined by $\lambda(t) =\begin{cases} 0, & \text{for } t \leq 0 \\ e^{-1/t}, & \text{for } t > 0. \end{cases}$ This is a smooth function with ...
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0 votes
2 answers
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Analytic continuation of $y=\sqrt{x^4-x^2+1}$

Question. Consider the function $y=\sqrt{x^4-x^2+1}$. Consider a sufficiently large circle $C$ of $0$ in $\mathbb{C}$, and let $x_0$ lies in this circle (suppose $x$ is not a branch point of this ...
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