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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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84 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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46 views

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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28 views

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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3answers
81 views

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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1answer
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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16 views

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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21 views

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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1answer
15 views

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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83 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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46 views

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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1answer
66 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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1answer
25 views

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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1answer
28 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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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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40 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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1answer
37 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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21 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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1answer
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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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19 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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2answers
51 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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24 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 $) , ...
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2answers
25 views

Show that a function is analytic on the punctured complex plane

I have to show that show that $\sum_{k = 1}^{\infty} \frac{1}{n! z^n}$ is analytic on $\mathbb{C}\{0\}$ and calculate its integral around the unit circle. My attempt is to try and use the analytic ...
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3answers
54 views

Show that $f(x) = 0$ if $x = 0$ $f(x) = \text{exp}(-1/x^{2})$ if $x \neq 0$ is infinitely differentiable but not analytic at $x_{0} = 0$ `

So taking the derivative of the latter half we get $$\left(\frac{2}{x^{3}}\right)e^{\frac{-1}{x^{2}}}$$ by chain rule. Im not sure how to find the derivative at x=0, and show that it equals 0 as it ...
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0answers
28 views

Properties of Complex Function $f(z)=e^{-\frac{1}{z^{1/3}}}$

This post will be about a part of an example from my complex analysis book. Problem: They claim that there exist a function $J_+(z)$ holomorphic in the upper half plane $\operatorname{im}z>0$, ...
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19 views

For $|q|<1$, the function $\frac{(az;q)_\infty}{(z;q)_\infty}$ is analytic on $|z|<1$.

I want to prove that for $|q|<1$, the function $f(z):=\frac{(az;q)_\infty}{(z;q)_\infty}$ is analytic on the set $\{z:|z|<1\}$. My approach: We consider the sequence of functions $\{f_n\}$ ...
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Prove that there is an analytic branch of $\sqrt{1-z^2}$ on any set having $\pm 1$ is in the same component in the complement

Show that a single valued analytic branch of $\sqrt{1-z^2}$ can be defined in any region $\omega$ such that $\pm 1$ are in the same component if the complement. Also, ehat are the possible values of $\...
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3answers
52 views

Finding all differentiable $f(z) = u(x) + iv(y)$ in $\mathbb{C}$ where $u(x),v(y)$ are real valued functions.

Finding all differentiable $f(z) = u(x) + iv(y)$ in $\mathbb{C}$ where $u(x),v(y)$ are real valued functions. I’m not sure what to do. Would $f$ be differentiable simply if and only if both $u$ and $...
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1answer
21 views

Express a complex analytic function as a single power series

Let $\Omega\subseteq\mathbb{C}$ be a connected open set, then $f:\Omega\rightarrow \mathbb{C} $ is analytic if for every $z_0\in\Omega$ there are $r>0$ and $(a_n)\subseteq\mathbb{C}$ such that $f(z)...
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1answer
24 views

Extending uniform convergence of analytic functions on larger domains

Let $f_k, f: ]-\infty , 1 [ \to \mathbb {R}$ be analytic functions. Suppose $f_k $ converges uniformly to $f $ on $]-\infty,0] $. Is it true that $f_k$ converges to $f$ on $]-\infty, \epsilon [$ for ...
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25 views

Proving that if $f=0$ on an arc or region in $D$, where $f$ is analytic on $D$, than $f = 0$ on $D$.

In the book of The Theory of Functions by Titchmarsh, at page 88, it is first stated that Let $f(z)$ be a function analytic in a region $D$, and let $P_1, P_2, > ...$ be a set of posts having a ...
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1answer
43 views

Domain of analyticity of $f(z) = \frac{1}{z^2}$

Let's say I have a function $f(z)$ = \begin{cases} 1/z^2&\text{if}\, z\neq 0\\ 1&\text{if}\, z= 0\\ \end{cases} And I want to find the domain in which this function is analytic, at first ...
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1answer
20 views

Using Cauchy Riemann complex form to prove Cauchy Riemann polar coordinates form

So I know that the regular Cauchy Riemann equations are: $\begin{equation} \frac{\partial u}{\partial x} =\frac{\partial v}{\partial y} \ \ \ \ \ \text{and} \ \ \ \ \frac{\partial u}{\partial y} =-\...
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1answer
23 views

Prove or disprove there is an analytic function $F$ on $\{ 1 < |z| < 2\}$ with $\Re F(x+iy) = \log (x^2+y^2)$

On the open set $U := \{ 1 < |z| < 2\} \subset \mathbb C$ the function $u(x,y) = \log(x^2+y^2)$ is $C^\infty$ and harmonic. Prove or disprove that there is an analytic function $F(z)$ on $U$ ...
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1answer
27 views

Analytic continuation of quotient of analytic functions

Suppose $f(z)$ and $g(z)$ are defined for some open subset $U$ of the complex plane, and that they are holomorphic on that subset. We then know that their pointwise quotient $(f/g)(z)$ is meromorphic ...
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53 views

Analytics function satisfied certain equality

Question: If $f(z)$ is analytic in a domain $D$, $f'(z)\neq 0$, $|f(z)|\neq 1$, then show that $$w=log\frac{|f'(z)|}{1-|f(z)|^2}$$ satisfy $\nabla^2w=4e^{2w}$. I don't know how to start with this ...
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1answer
63 views

Smooth and not analytic

Can someone show me, without reference to Taylor series, why a complex function can be smooth but not analytic? I do not understand it intuitively or visually either. I would like an explanation ...
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23 views

An understanding problem of analytic function

I am reading a textbook of distribution theory. Here is the definition of test functions: The class of test functions $\mathcal D(\Omega)$ consists of all functions $\varphi(x)$ defined in $\Omega$, ...
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20 views

Can the following equation be solved analytically?

Is it possible to solve the following equation analytically for r: $$A = exp\bigg[a\bigg(\frac{r}{r_{0a}}\bigg)^b\bigg] + exp\bigg[c\bigg(\frac{r}{r_{0c}}\bigg)^d\bigg] + exp\bigg[f\bigg(\frac{r}{r_{...
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2answers
26 views

Does this equation have any analytical solution?

I am trying to compare my numerically calculated model with analytical solution, which should be provided by following equation $f(x) = e^{\frac{[f(x)]^2}{2}+ax+b}$ where $a$ and $b$ are constant. ...
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2answers
94 views

Find the zeros of $f(z)=z^3-\sin^3z$

I want to find the zeros of $f(z)$, $$f(z)=z^3-\sin^3z$$ My attempt $f(z)=0$ $z^3-(z-z^3/3!+z^5/5!-\dots)^3=0$ $z^3-z^3(1-z^2/3!+z^4/5!-\dots)^3=0$ $z^3[1-(1-z^2/3!+z^4/5!-\dots)^3]=0$ So $z=0$ ...
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1answer
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Fixed points of complex analytic functions

We have given some complex analytic function $f:\mathbb{C} \to \mathbb{C}$. I have read in a proof that if $f$ has exactly one fixed point $v \in \mathbb{C}$, then follows that $f$ is a polynomial ...
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1answer
22 views

Existence of convergent sequence in domain

Im looking at a proof where there are two analytic functions $f,g$ where $g$ is injective and maps a domain onto the open unit disk $\mathbb{D}$ and $f$ maps the same domain onto a domain $G$ with $\...
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3answers
60 views

Analytic function is identically zero on open unit disk if $|f(z)|\leq 1-|z|$. [duplicate]

Let $f$ be an analytic function defined on open unit disk with $|f(z)|\leq 1-|z|$. I need to establish $f$ is zero on disk . If I'm able to prove that $f(0)=0$ and $f(0)\leq f(z)$ then $f$ is ...
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1answer
39 views

If $f(z)$ is analytic inside and on the circle $|z| = 2$, is $\oint_{|z|=2}f(z) = 0?$

I know if $f$ is analytic in a simply connected domain D then for any closed loop in D this integral is 0. Is that enough to say yes to this question?
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1answer
22 views

Question about proof that $Ln$ is analytic.

I have a question about a step in the proof of the following theorem. Theorem: $Ln: \mathbb{C}- \{x \in \mathbb{R}: x \leq 0\} \to \mathbb{C}: z = re^{i \phi} \mapsto \ln r + i \phi$ $(\phi \in (-\pi,...
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0answers
23 views

Contour Deformation in the Laplace Inversion Formula

Following Szpankowski - Average Case Analysis of Algorithms on Sequences the exponential generating function of $g(n)$ which is thought to be analytic in $n$ is defined as $$ G(z)=\sum_{n=0}^{\infty} ...
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2answers
57 views

Prove that there exists no non-zero analytic function such that $f(\frac{1}{n})$=0 for all n

There exists no non-zero analytic function such that $f(\frac{1}{n})$=0 for all n Im trying to prove by differentiability If $f$ is differentiable it is continuous at $0$, So $f(0) = \lim f(\frac{1}...
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1answer
44 views

A query while showing that the Gamma function $\Gamma$ is logarithmically convex for $x \gt 0.$

We are using the general definition of gamma function defined on $\Bbb C \setminus \{\text{non-positive integers}\}$ i.e. $\Gamma(Z)=\frac {e^{-\gamma z}}{z} \prod (1+ \frac zn)^{-1} e^{\frac zn}$. ...
1
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1answer
67 views

Is there any difference between the elements in $\mathbb{C}[x]$ and $C^{\omega}$?

Studying Hilbert spaces, I've been told that $\{1, x, x^2,\dots, x^k,\dots\}$ form a basis (EDIT: when orthonormalised) in $\mathcal{L}^2([a,b])$ because the set of all polynomial functions is dense ...