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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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If v is harmonic conjugate of u, then the harmonic conjugate of $3u^2 − v^3$ is harmonic conjugate of

Attempt: I tried to use cr-equations,but integration becomes clumsy. Also, I tried to form another holomorphic function which have real part equal to $3u^2 − v^3$ using any other holomorphic functions,...
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$\int_0^{+\infty} f(x) dx = - \lim_{a\rightarrow 0^+} \frac{1}{a} \sum Res_{\Bbb{C} \setminus \Bbb{R}^+} (f(z)z^a)$?

I find integrals on the real axes computed by complex number techniques and looking for a generalization working just on the positive semiaxes. I tried to put together this general result, but I am ...
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Confusion about relationship between continuous and real analytic functions

By Weierstraß aprroximation theorem, every continuous function can be $\epsilon$-approximated by a polynomial, i.e., if $f:[a,b]\to\mathbb{R}$ is a continuous function, then for all $\epsilon>0$ ...
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What's the product of Bergman and Dirichlet functions on the disk?

Under what conditions can we claim that the product of a Bergman space function and a Dirichlet space function belongs to the Hardy space $H^2$ (or any $H^p$ with $p>0$), all on the unit disk $\...
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Convergence of Power Series and Power Series Solutions of ODE

A function which has a convergent power series expansion about a point is called analytic at that point. A function may not be analytic at some points but analytic every where else. This means that ...
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69 views

Show that f is constant. [duplicate]

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an analytic function and let $a \in \mathbb{R}$. Show that if $\Re f(z) \geq a$ for all $z \in \mathbb{C}$, then $f$ is constant. Now I know that I need ...
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21 views

Some questions regarding complex analysis

I recently came across the following problem Pick out the true statements: There is a bijective analytic function from $\Bbb C$ to the upper half plane $\Bbb H$ There is a non-constant ...
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Connectedness of the domain in Identity Theorem

Let $f$ be a complex-valued holomorphic function defined on an open set $\Omega\subseteq\mathbb{C}$, $f:\Omega\rightarrow\mathbb{C}$, which is not identically zero. Let $S=\{a\in \Omega :f(a)=0\}$=...
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Complex Analysis, find all analytic functions

Find all analytic functions $f: \mathbb{C} \rightarrow \mathbb{C}$ such that $|f(z)-1| + |f(z)+1| =4 $ for all $z \in \mathbb{C} $ and $f(0) = \sqrt{3} i$ I understand that the given equation ...
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75 views

Does anyone knows how to average function in three dimensions

$$ \begin{split} \frac{dr}{dt} &= -\epsilon \sin^2(\theta)z \\ \frac{d \theta}{dt} &= -1-\epsilon \cos\theta \sin\theta z\\ \frac{dz}{dt} &= \epsilon(r^2-T) \end{split} $$ ...
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Integral of real part of $z$ around the unit circle

What is the result of integrating the real part of z (a complex number) anti clockwise around the unit circle? At first glance, I couldn't identify any points within the circle where analyticity ...
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How to justify this statement?

Let $f$ a holomorphic function in $\mathbb{D}(0,1)$ such that $f(z)=\sum_{n\ge 0}a_n z^n$. Then apparentely we can deduce that for all $r\in [0,1[$, $n\ge1$ : $\int \limits_{0}^{2\pi} \overline{f(r\...
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Analytic contiunation

this is more of a broader question. Say I have an analytic complex function $f$ that is defined on the open unit circle, but I know that the limit of $f$ when $|z|$ approaches 1 is 0. Can you define ...
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Is there a “monotonicity” property for analytic continuation?

If I have two complex functions defined by power series $A(z) = \sum a_n z^n $, $B(z) = \sum b_n z^n$ with $|a_n| \ge |b_n|$ for all $n$, and I know that $A$ converges in some set $U_1$ and defines a ...
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37 views

Different ( Is it equivalent?) hypothesis for stronger version of Goursat's theorem

Ahlfors, states that Theorem: Let $f(z)$ be analytic on the set $R'$ obtained from a rectangle $R$ (interior and on the boundary of the rectangle) by omitting a finite number of interior points $a_i$...
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Existence of the limit of a bounded analytic function

Let $X$ be a Banach space and let $U\subset\mathbb{C} $ be open. If we have an analytic function $f:U\setminus \left\{ 0\right\} \rightarrow X$ such that $\underset{x\in U\setminus \left\{ 0\right\} }{...
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38 views

How is it possibile for a complex analytic function not having a derivative in the region in which it is analytic?

There is a question asking if the function $f$, which is defined by $$f(z) = \frac{\bar{z}^2}{z},\quad z\neq 0$$ and $f(0) = 0$, is analytic at $z = 0$ and if there is $f'(0)$. The answer is ...
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113 views

$f(x+y) = f(x) + f(y) + f^k(x) f(y) + f^k(y)f(x) $

Let $k>1$ be an integer. Consider equations of type $$f_k(x+y) = f_k(x) + f_k(y) + f_k^k(x) f_k(y) + f_k^k(y) f_k(x) , f(-z) = f(z), $$ where $f_k$ is the $k$ th function and $f_k^k $ is the $k$...
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Is there a name for the square of a function plus the square of its Hilbert transform?

Given a real-valued analytic function $f$ defined on the whole real line, and its Hilbert transform ${\cal H}f$, it seems that the quantity $f(x)^2+{\cal H}f(x)^2$ should have some kind of importance ...
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$f$ has a pole at $z=a$ implies $1/f$ has a removable singularity at $z=a$

In Section V.1 of Conway's Functions of One Complex variable, he says that if $f$ has a pole at $z=a$ implies $[f(z)]^{-1}$ has a removable singularity at $z=a$. I am confused why $[f(z)]^{-1}$ should ...
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$f$ is analytic in the unit disk such that $|f(1/n)|\leq e^{-\sqrt{n}}$. Prove that $f$ Is identically zero.

$f(z)$ is analytic in the unit disk such that $|f(1/n)|\leq e^{-\sqrt{n}}$ for $n=2,3,...$ Prove that $f$ Is identically equal to zero. My Attempt: Consider the Taylor expansion $$f(z)=f(0)+f'(0)+\...
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General formula for integrating analytic function with linear term

Given an analytic function $\, f : \mathbb{R} \rightarrow \mathbb{R}$ with no known closed formula for its antiderivative. Assume further that by some clever tricks you managed to calculate the ...
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26 views

what is the mapping of horizontal lines and vertical lines under $w(z)=\cos(z)$ in general? [duplicate]

For the mapping of horizontal lines ($z=x+iy_0$ for fixed $y_0$) and vertical lines ($z=x_0+iy$ for fixed $x_0$) under $w(z)=\cos(z)$, are there any general formulas. What I mean is, is there a ...
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1answer
65 views

what is the mapping of horizontal lines and vertical lines under $w(z)=sin(z)$ in general?

For the mapping of horizontal lines ($z=x+iy_0$ for fixed $y_0$) and vertical lines ($z=x_0+iy$ for fixed $x_0$) under $w(z) = \sin(z)$, are there any general formulas? What I mean is, is there a ...
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25 views

Why is “If f is a function such that both f and the conjugate of f are analytic on a domain D, then f is constant.” true? [duplicate]

I tried writing f as f=u+iv and conjugate of f=u-iv, but I don't know how to apply this to prove the statement. Do we need to apply the Cauchy-Riemann equations for this? How can we prove that f'=0 if ...
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When to resolve into partial fractions for applying Cauchy's integral formula?

Suppose I have to calculate $\int \frac{f(z)}{(z-a)(z-b)}dz$ around a curve in which both $a$ and $b$ lie inside. Should I apply Residue theorem like this: Residue at $a$= $\frac{f(a)}{a-b}$ ...
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Definition of analytic function at a point

Suppose $f:I\to \mathbb{R}$, where $I$ is an open subset of $\mathbb{R}$, is a smooth function on $I$, $f\in C^{\infty}(I)$. Let $x_0\in I$. Def. We say that $f$ is analytic on $x_0$ if the Taylor ...
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Is $f_{n}$ is analytic on $(a, b)$ and $f_{n} \rightarrow f$ uniformly on $(a, b)$ then is $f$ analytic on $(a, b)$?

Is $f_{n}$ is analytic on $(a, b)$ and $f_{n} \rightarrow f$ uniformly on $(a, b)$ then is $f$ analytic on $(a, b)$? Intuitively, I think that the answer is no. I know that the statement holds for ...
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91 views

If $Re(f)$ is a polynomial then $f$ is a polynomial, where$f$ is entire

If $f$ is entire, and $Re(f)$ is a polynomial in x,y I am trying to show that f is a polynomial in z. We can write $f(z) = u(z) + iv(z)$. where $Re(f) = u(z)$ I have solved problems similar to this ...
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Textbook Proposition on Product of Real Analytic Functions

Let \begin{align*} \sum\limits_{j=0}^\infty a_j (x-c)^j && \sum\limits_{j=0}^\infty b_j (x-c)^j \\ \end{align*} be two power series with intervals of convergence $\mathcal{C}_1$ and $\...
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Textbook Proposition on Quotient of Real Analytic Functions

Let $f$ and $g$ be real analytic functions, both of which are defined on an open interval $I$. Assume that $g$ does not vanish on $I$. Then the function \begin{align*} h(x) &= \frac{f(x)}{g(...
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Proof that, if $f$ and $|f|$ are analytic, then $f$ is constant

I am trying to show that assuming for an analytic $f$ we have $|f|$ is also analytic. That the original f must be constant. My original thought was to use the Cauchy Riemann equations to try and show ...
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50 views

If $f$ is $\mathbb R^2$-differentiable and the limit $\lim\limits_{h \to 0} \Re\left(\frac{f(z+h)-f(z)}{h}\right)$ exists, then $f$ is holomorphic?

Assume $f$ is defined on some domain $D\subset\mathbb C$, differentiable on $D$ as a function of two real variables, and that, for all $z\in D$, the limit $\lim\limits_{h \to 0} \Re\left(\frac{f(z+h)-...
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Cauchy-Riemann equations for $z=x+iy$ and $f(z)=R(x,y)e^{i\theta(x,y)}$

I know the following forms of the Cauchy-Riemann equations (please correct me if I made a mistake): $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$, $\frac{\partial u}{\partial y} = -...
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Is there a continous branch in the neighbourhood of 0

Is there a continuous branch in the neighborhood of function $z$ around $0.$ In my opinion no, because as we take a circle and go around it, the argument gains $2\pi$ and the value of logarithm ...
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Does f(z) have a continous branch of logarithm on this set

As stated in the question, the set $A$ is $z: 0<|z|<3$ and $f(z)=\frac{z}{z+3}$ I know it does not, it is easy to show by taking derivative of $Logf(z)$ and calculating the integral on circle (0,...
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124 views

Analytic functions having harmonic real and imaginary parts.

I've bee set the following question in a homework assignment for my complex analysis class, but have literally no idea what it means by sufficiently regular. Let $f : \mathbb{C} \to \mathbb{C}$ be an ...
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Singularity of $\frac{z+2}{e^{\frac{1}{(z+2)^2}}}$

I have a singularity in $z=-2$, now I wolud like to find the kind of singularity, so I have to compute the limit: $$\lim_{z\to-2}\frac{z+2}{e^{\frac{1}{(z+2)^2}}}$$ for me this limit is $0$, because ...
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Why is $\frac{\partial \bar f }{\partial z} = 0$ for an analytic function $f$?

Why is $\frac{\partial \bar f }{\partial z} = 0$ for an analytic function $f$? I understand that, for an analytic function $f, \frac{\partial f }{\partial \bar z} = 0$, but I do not get why the above ...
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69 views

$f$ is an entire function s.t $|f(z)|=1$ $\forall z \in \Bbb R$. prove that $f$ has no zeros in $\Bbb C$

My attempt: I was trying to apply identity theorem that if the zeros of the function do have any limit point and it will be a zero function but setting $g(z)=f(z)-1$ will not help me. Can anyone help ...
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Why is it that for a hamonic $u$, $\int_{\gamma}*du = 0$ for any cycle $\gamma$ then $u$ has a harmonic function there?

Let $u$ be a harmonic function on a connected open set. If $\int_{\gamma}*du = 0$ for any cycle $\gamma$ then $u$ has a harmonic function. This question arises from an answer to this post Please do ...
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33 views

Complex line integrals

Suppose we have an analytic function then Why complex integral of that function does not depend on the path of integration?
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66 views

Is the Lambert W function analytic? If not everywhere then on what set is it analytic?

I would appreciate if someone can help me answer the following questions. Although I read several papers and documents on the Lambert W function, I could not assess on what set is this function (or ...
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1answer
49 views

A function is 0 if is almost bounded

I am having problems with this excercise, if $f$ is analytic in $D \subset \mathbb{C}$ and: $$|f(\cos(z))| \leq m|z|^n$$ for some $m,n \in \mathbb{N}$ and $z \in D$. Then show that $f \equiv 0$. ...
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39 views

How to solve the recursive equation $y^{(n+2)} +(n+1)y^{(n+1)} +\tfrac{n(n+1)}{2} y^{(n)}=0$

I encounter the problem when I try to get the Taylor series of $\arctan x$ at $x=1$. It seems that the method for expanding it at $x=0$ does not work anymore (which is obtained by observing that $\...
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27 views

Closed curve for an analityc function

If I have an analityc function $f(z)$ on a domain $B \subset \mathbb{C}$ and a simple and closed curve $C$ that encloses $B$, and if $|f|$ is constant over C, the $f$ is constant on $B$? I haven’t ...
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21 views

Properties of analytic functions

I read some chapters of Penrose book "Road to Reality". It seems that analytic functions are one of the more important concepts of the book. They are listed many properties: 1) Analytic functions ...
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24 views

A function analytic in the unit disk belongs to the class Nevanlinna if and only if it is the quotient of two bounded analytic functions

I'm trying to understand a part of this proof from Duren, in the converse, I don't see it clear when it says "by analytic completion of the Poisson Formula,..." and then the result; I tried to prove ...
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53 views

Show $ \int_{0}^{1}\frac{\sin (ax)}{x}dx $ is an entire function.

I am struggling in evaluating the following integral: $$\int_{0}^{1}\frac{\sin (ax)}{x}dx$$ I know that if the integral is from $0$ to infinity, it will be a constant of $\pi/2$ which is analytic, ...
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25 views

Mapping and Cauchy- Reimann conditions

If a complex function is analytic it must hold Cauchy-Reimann Conditions, So conjugate of F(Z) is irrotational and solenoidal , Does this points mapping from $R^2 $ to a subspace of $R^2 $ ($R^1 $) , ...