Questions tagged [analyticity]

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 of functions over the complex numbers.

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Asymptotic expansion of non analytical function

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a real smooth (not necessarily analytical) function. Suppose I tell you that $f(t)$ admits a full asymptotic, at $t\rightarrow0$ expansion up to all orders ...
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Real-analytic ODE solution with respect to parameter

Consider the following parameter-dependent ODE: \begin{align} & x'(t;p) = f(t,x(t;p);p), \qquad t \in (0,T), p \in P \\ & x(0;p) = x_0 \end{align} where $f \colon \mathbb{R} \times \mathbb{R} \...
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What is an analytic space?

What is an analytics space? I have seen more definitions, so my question is - what is the version used the most? And is it all the same notion, or can there be confusion regarding more terms with the ...
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Notation of analytic functions $\mathcal{C}^{\omega}(\mathbb{R}^n)$

I understand that $\mathcal{C}^{\omega}(\mathbb{R}^n)$ is the notation for analytic functions on $\mathbb{R}^n$. In what books can I find this notation?
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Analytic functions at points other than zero

I am confused about the definition of analytical functions on real line, or, to be more precise, how this definition is used. According to Wikipedia Analytic function is a function that is locally ...
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A generic complex contour integral

If, let's say, I have an integral such as $$\int_{-\infty}^{\infty}\frac{dt}{(t-a)(t-b)(t-c)(t-d)}$$ where $a$, $b$, $c$ and $d$ are complex numbers such that the poles falls in all the four quadrant ...
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Find extrema of $y=?(x)-x$ with the Minkowski Question Mark function

The Goal: is to figure out the global extrema of the Minkowski Question Mark function $?(x)$. Here is the graph of: $$?(x)-x:$$ The $y$ value of the global maximum was found by systematically ...
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Stuck proving that $\mid z\mid^2$ is not analytic in $z$

I am proving that $f(z) = \mid z\mid^2$ is not an analytic function. So i didn't want to use the Cauchy-Riemann condition or anything but i know that this particular function is diffentiable at only $...
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Do real-analytic functions that uniformly converge in all derivatives make something good?

Given a sequence of real-analytic functions $$f_n:(-\varepsilon_n,1+\varepsilon_n)\to\mathbb{R}$$ and assume that each of their derivatives converges uniformly on $[0,1]$: $$\forall k=0,1,\ldots,\...
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Proving that : $Q_0(z)=\frac{1}{2}\ln\left(\frac{z+1}{z-1}\right)$ is single valued.

The Legendre function of second kind $Q_\nu(z)$ has branch points at $z=\pm 1$. The branch points are joined by a cut along the real axis. Show that $$Q_0(z)=\frac{1}{2}\ln\left(\frac{z+1}{z-1}\right)$...
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real analyticity of harmonic function

I am working on an exercise, the teacher has given the answer to the exercise, but I can’t understand it. Exercise Let $U$ be an open set of $\mathbb{C}$, $u: U\to \mathbb{R}$ be a harmonic function, $...
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Proving whether a complex function is analytical help!

I am currently struggling to prove whether a complex function is analytical. I understand that I must employ the Cauchy-Riemann relations to do this. However, the answer I get is one that I can't ...
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Suppose that $f(z) = \frac{z}{1-z}$. Use Cauchy-Riemann Equations to determine the analyticity of $f(z)$. Then find the derivative of $f(z)$. [closed]

can anyone help me in solving this question? I have tried expressing $z=x+iy$ and multiplying the conjugate to numerator and denominator. However, the expansion and differentiation are too tedious. Is ...
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Holomorphic is much stronger than real differentiable in a neighbourhood. What about analytic?

For real of 1 variable: Analytic $\implies$ smooth $\implies$ (real-)holomorphic $\implies$ differentiable (at a point $p$). here, real holomorphic at $p$ just means real differentiable in an open ...
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If $f$ is analytic on an open and simply-connected subset of $\Bbb{C}$, and if $\cos{f}$ is constant, then must $f$ also be constant?

I need help with the following question: Suppose $f$ is analytic on the open and simply connected space $\Omega\subset\mathbb{C}$. Must $f$ be a constant function if $\cos{f}$ is constant? To me it ...
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Domain of $I(z) = \int_0^\infty \frac{u^{z-1}}{1+u} du$, where z is complex and is it analytic?

Consider the function $I(z)$ defined through the integral $I(z) = \int_0^\infty \frac{u^{z-1}}{1+u} du$, where z is a complex parameter, what is the maximal choice for the domain of the function and ...
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Is $f(r \cos \theta, r \sin \theta)=\theta$ real-analytic?

Let $U=\{(x,y) \in \Bbb R^2: y>0\}$ and $f:U \to \Bbb R$ defined by, $f(r \cos \theta, r \sin \theta)=\theta$. Is this function real-analytic? I know that polynomials in usual $x,y$ coordinates ...
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extension problem with motivation

I am motivated by the Lorenz curve used in economics and statistics - a proper multivariate generalization will help researchers (Arnold, Taguchi, etc.) assess multidimensional risk and inequality (...
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Existence of Analytic and non constant function

Can we construct a non constant analytic function $f(z)$ on open unit disc such that: $\frac{1}{\sqrt{n}} \lt |f(\frac{1}{n})| \lt \frac{2}{\sqrt{n}} \forall n \in \mathbb N $ My approach: I think ...
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Cauchy-Riemann equations and required continuity of derivatives

So I just read, that for any analytic function, the Cauchy-Riemann equations will hold. However, the reverse, i.e. Cauchy-Riemann equations hold -> function is analytic, is supposedly only true if ...
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Needs hints or help for a complex analysis question

I am having problem with the following question. Have: $f(z)$ be analytic on an open set $\Omega$ with $\overline{\Delta(0, r)} \subset \Omega$. for zeros of $f$ in $\Delta(0, r)$ are $a_{1}, a_{2}, \...
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Can you always specify the value of a $C^{\infty}$ or analytic function on an isolated set? [duplicate]

Former math grad student, now a lawyer for the last $28$ years. Just doing math for fun in my spare time. I was browsing questions here and my mind went on a tangent. The following questions ...
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Is $ z / (e^z - 1) $ analytic in $ |z| < \pi $?

I see that $ z / (e^z - 1) $ has a singularity at $ z = 0 $. Is $ z / (e^z - 1) $ analytic everywhere in $ |z| < \pi $? If yes, how is it determined to be analytic everywhere despite a singularity ...
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Question on the definition of analytic continuation along a path

In my notes I have that $(g,E)$ is an analytic continuation of $(f,D)$ along a path $\gamma :[0,1]\to U$ if there exists function elements $(f_i,D_i)$ with $i\in \{ 0,\ldots ,n\}$ and $0=t_0<t_1<...
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Power series of a function which is analytic everywhere

I am having doubt regarding the power series of the function $g(x) = \frac{1}{1+x^2}$. Since the function $f(x) = {1+x^2}$ is analytic everywhere and by using the theorem that if $f(x)$ is analytic ...
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What will happen if the "connectedness" is dropped/discarded from the domain of analytic function?

Let $G\subset\mathbb C$ be a non-empty open connected set and the function $f:G\to \mathbb C$ be analytic. It is easy to show that if there is a point $a\in G$ such that $f^{(n)}(a)=0$ for all $n\geq ...
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Analytic dependence of an improper integral on parameter

I want to consider the integral $$ \int_0 ^\infty \frac{x^{\alpha}}{(1+2xt+x^2)^\beta}dx; \quad -1<t<1.$$ We can assume that $\alpha,\beta$ are such that it converges. I suspect that this ...
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Relationship between analytic and square-integrable properties of a function

Schutz's book (p.10) mentions in passing a theorem in functional analysis that says that any square-integrable function $g$ may be approximated by an analytic function $f$ in such a way that the ...
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Formulation of the theorem stating that an analytic function around $z_0$ with $f(z_0) = 0$ is not identically zero in some neighbourhood of $z_0$.

In my complex analysis textbook the following theorem is stated: Theorem. Given a function $f$ and a point $z_0$, suppose that (a) $f$ is analytic at $z_0$; (b) $f(z_0) = 0$ but $f(z)$ is not ...
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Definition of local and global analytic isom between Riemann surfaces

I have question about two technical term of Riemann surfaces. Let $X$ and $Y$ be Riemann surfaces. My understanding is that local analytic isomorphism at $p$ between $X$ and $Y$ is defined as analytic ...
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Proof in complex function - Is the function bounded?

Let $f: \mathbb{C} \to \mathbb{C}$ be an entire function which is not constant. $g: \mathbb{C} \to \mathbb{C}$ is defined as $$ g(z) =\begin{cases} 1, & f(z)=0 \\[2ex] \dfrac{|f(z)|+1}{|f(z)|} \...
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Non-existence of a holomorphic map on the disc

Is there a function $f$ that is holomorphic on the punctured unit disc such that $f^{\prime}$ has a pole of order $1$? My answer is: No. The following post has an answer to my question. Show that ...
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How is this argument in this paper justifiable?

Eq. 10 in this paper https://arxiv.org/pdf/1806.04039.pdf has an integral equation \begin{equation} \sigma(\lambda)=a_1(\lambda)+\frac{1}{N}a_1\left(\lambda+\frac{1}{g}\right)-\int_{\mathcal C} d\...
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If both $f(z)$ and $−\overline{f(z)}$ are analytic, what can you say about $f(z)$? Prove your claim.

I am struggling with this question. If both $f(z)$ and its negative conjugate $-\overline{f(z)}$ are analytic, what can you say about $f(z)$? Prove your claim. I know that the sum on difference of two ...
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Taylor series generated by a function

There are questions related to the current question like in this thread or in this thread, but I do have something specific to ask. If this indeed is a duplicate, then please let me know the thread ...
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Why does rational dependence of $f'$ on $f$ imply that $f$ is real-analytic?

Let $f:(0,1) \to \mathbb{R}$ be a smooth function, satisfying the ODE $$ f'=\frac{P(f)}{Q(f)}, $$ for some polynomials $P,Q$. We assume that $Q(f)(x)=Q(f(x))$ does not vanish on $(0,1)$. How to prove ...
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Analytic metric on Riemannian manifold

What is meant for a metric being analytic on a Riemannian manifold? I am looking for the definition, and if you could write some bibliography where I can read about analyticity on riemannian metrics ...
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Riemann zeta function is analytic for $\operatorname{Re}(z)>1$

I want to show that, Riemann zeta function is analytic on the domain $\operatorname{Re}(z)>1$. I know that it is absolutely and uniformly convergent on the right half of the line $Re(z)=1$. Now, I ...
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Analyticity of a complex function..

A paper I am reading says that $e^{-i q \left(-\log (-q+i 0^+ )+1+\frac{i \pi }{2}\right)}$ is a function analytic on upper half plane and $e^{-i q \left(-\log (q+i 0^+ )+1+\frac{i \pi }{2}\right)}$ ...
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Sequential approach to differentiability of complex valued function

Let $f$ be analytic function on an open set $G$. Let $z\in G$. Let $\{z_n\}$ and $\{w_n\}$ be two sequences in $G$ which converges to $z$. Then show that $lim_{n \to \infty} \frac{f(z_n)-f(w_n)}{z_n-...
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Unsure about a step in the analyticity of a holomorphic function

Consider the following section in a proof in my notes: We show here that, as a consequence of Cauchy's integral formula any holomorphic function g(z) in an open domain D is analytic. Consider a circle ...
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What is an analytic coefficient? (Kashiwara)

In Kashiwara's thesis he uses $\mathcal{D}$-modules to investigate (systems of) linear PDEs with analytic coefficients. There are also mentions of analytic manifolds analytic linear PDEs analytic $\...
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Conjecture on openness of an analytic mapping

Consider a real analytic function $g: \mathbb{R}^m \rightarrow \mathfrak{M}_{\mathbb{R}}(n, k)$ to the set of $n \times k$ matrices where $k \ge n + 1$ such that $\forall x \in \mathbb{R}^m \quad \...
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Analytic region of the function $z^{2 \alpha}$

This is a paragraph from Ahlfors Complex Analysis It states that for the function $z^{2 \alpha}$ where $0<\alpha<1$, it is possible to choose an analytical branch of the function whose argument ...
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Bolzano-Weierstrass and zeros of complex analytic function

I am working on a textbook exercise. A similar question: An analytic function in a compact region has finitely many zeros, but it's not quite clear to me and I also have possibly another approach? I ...
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Real-valued analytic function (complex)

I am studying elementary complex variable topic. An exercise in Brown-Churchill book asks us to show that if $f$ is analytic in a domain $D$ and real-valued at all points then $f$ must be constant ...
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Asymptotics in generating functions, am I doing right?

For each generating function $A(x)$, find c and ρ such that the coefficient $a_n$ of $x^n$ in the generating function $A(x)$ satisfy: $a_n \sim c * ρ^n$. $$A(x)= \frac{1-2x}{(1-x)(1-3x)}$$ $$A(x)=\...
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5 votes
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The coefficient and asymptotic in generating function

Let $L_n$ be the set of all the paths from $(0,0)$ to $(n,0)$ such that every step is $u=(1,1)$ , $d=(1,-1)$ and $r=(2,0)$. Notice that the path could go under the $x$ axis. a. Write a generating ...
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If $\sum_{n=1}^{\infty} 1/a_n $ diverges, can $\prod_{n=1}^{\infty} (1-z/a_n) $ be analytic anywhere

This question was asked in my complex analysis assignment. Question :If $\sum_{n=1}^{\infty} 1/a_n $ diverges, can $\prod_{n=1}^{\infty} (1-z/a_n) $ be analytic anywhere? I think it can't be as we ...
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On the class of real-valued functions which extensible to entire functions

Let $\mathcal{D}$ be the class of real-valued functions $f(x)$ defined on $\mathbb{R}$, which are extensible to entire functions $\bar{f}(z)$ on $\mathbb{C}$ such that $\bar{f}(x)=f(x)$, for all $x \...
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