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 ...

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21
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1answer
402 views

Is it possible for a function to be smooth everywhere, analytic nowhere, yet Taylor series at any point converges in a nonzero radius?

It is well-known that the function $$f(x) = \begin{cases} e^{-1/x^2}, \mbox{if } x \ne 0 \\ 0, \mbox{if } x = 0\end{cases}$$ is smooth everywhere, yet not analytic at $x = 0$. In particular, its ...
18
votes
2answers
530 views

Function $f(x)=\int_0^\infty\left|\sin(t)\cdot\sin(x\,t)\cdot e^{-t}\right|\,dt$

Let $$f(x)=\int_0^\infty\Big|\sin(t)\cdot\sin(x\,t)\cdot e^{-t}\Big|\,dt,$$ where $|\dots|$ denotes the absolute value. We are concerned only with positive values of $x$ (i.e. let the domain of the ...
14
votes
3answers
288 views

What is the Riemann surface of $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$?

The following appears as the second-to-last problem of Stewart's Complex Analysis: Describe the Riemann surface of the function $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$. This problem ...
14
votes
3answers
432 views

Can a function “grow too fast” to be real analytic?

Does there exist a continuous function $\: f : \mathbf{R} \to \mathbf{R} \:$ such that for all real analytic functions $\: g : \mathbf{R} \to \mathbf{R} \:$, for all real numbers $x$, there exists ...
13
votes
1answer
405 views

Hilbert's 19th problem: Why do we care?

Hilbert's 19th problem asks: Are the solutions of regular problems in the calculus of variations always necessarily analytic? This was proven to be true (through the work of Sergei Bernstein, ...
11
votes
2answers
335 views

Holomorphic function on the unit disk $f$, show the set $z,w\in \mathbb{C}$ such that $f(z)=f(w)$ is not countable

Here is the problem statement: Suppose $f$ is a holomorphic function on the unit disk. Show that the set $A=\lbrace (z,w) \in \mathbb{C}^2\;|\; |z|,|w| \leq \frac{1}{2}, z\neq w, f(z)=f(w)\rbrace$ is ...
11
votes
1answer
309 views

Images of compact subsets in the plane

Let $K$ be an infinite compact subset of $\mathbb{C}$. Is it true that there exists a sequence $(f_n)_{n>0}$ of functions holomorphic in some neighborhood of $K$, such that the images $f_n(K)$ are ...
10
votes
2answers
123 views

Does $f'$ analytic imply $f$ analytic?

If $f'$ is known to be analytic, does it mean that $f$ is analytic as well? I've tried to expand $f$ and then to replace the tail of it by the expansion of $f'$, yet the factorials don't add up. I ...
8
votes
4answers
211 views

Real-analytic $f(z)=f(\sqrt z) + f(-\sqrt z)$?

Are there nonconstant real-analytic functions $f(z)$ such that $$ f(z)=f(\sqrt z) + f(-\sqrt z)$$ is satisfied near the real line ? Also can such functions be entire ? And/Or can they be periodic ...
8
votes
3answers
393 views

Smooth Pac-Man Curve?

Idle curiosity and a basic understanding of the last example here led me to this polar curve: $$r(\theta) = \exp\left(10\frac{|2\theta|-1-||2\theta|-1|}{|2\theta|}\right)\qquad\theta\in(-\pi,\pi]$$ ...
7
votes
4answers
953 views

How many smooth functions are non-analytic?

We know from example that not all smooth (infinitely differentiable) functions are analytic (equal to their Taylor expansion at all points). However, the examples on the linked page seem rather ...
7
votes
2answers
513 views

True/False Questions for Complex Analysis

I am studying for my (introductory) complex analysis final exam tomorrow. I am practicing an old final exam, which unfortunately has no answer key. Here is a link: ...
7
votes
1answer
137 views

Proving that a function is analytic

I'm struggling with the following problem: Problem: Suppose that $h$ is a continuous function on a simple closed curve $\gamma$. Define $$ H(w) = \oint_{\gamma} \frac{h(z)}{z - w} \, dz. $$ Show ...
7
votes
1answer
102 views

$\forall x \,\exists k$ s.t. $f^{(k)}(x)=0$, then $f$ is a polynomial

My friend sent me the following problem: Suppose that $f$ is real analytic on $(a,b)$, and that for all $x$ in $(a,b)$ there exists a non-negative integer $k$ such that $f^{(k)}(x)=0$. Show ...
6
votes
3answers
869 views

Explanation of Maclaurin Series of $x^\pi$

I am reviewing Calc $2$ material and I came across a problem which asked me to explain why $x^\pi$ does not have a Taylor Series expansion around $x=0$. To me it seems that it would have an expansion ...
6
votes
3answers
373 views

Expressing the area of the image of a holomorphic function by the coefficients of its expansion

I have the following problem. Let $f:D\to \mathbb C$ be a holomorphic function, where $D=\{z:|z|\leq 1\}.$ Let $$f(z)=\sum_{n=0}^\infty c_nz^n.$$ Let $l_2(A)$ denote the Lebesgue measure of a set ...
6
votes
1answer
339 views

Liouville's theorem problem

Hi i need some hints and help with this problem. Let $f\in\mathcal O(\mathbb C)$ and assume that $\Re f(z)\geq M$ for all $z\in\mathbb C$. Use Liouville´s theorem to prove that $f$ is constant ...
6
votes
4answers
575 views

composition of power series

Does anyone know how to derive a formula for the coefficients. That is if, $f(x)=\sum _{n=0}^{\infty } a_nx^n$ and $g(x)=\sum _{n=0}^{\infty } b_nx^n$ suppose the compostion is an analytic ...
6
votes
1answer
243 views

Extended Proof of the Theorem that a bounded analytic function is constant.

I am having trouble feeling convinced by my proof and more importantly - feeling confident in my working out. The question reads (a) Let $f$ be an entire function such that there exist real ...
6
votes
1answer
354 views

Complex differentiable but not analytic on circle of convergence

I'm trying to get a better handle on behavior of complex power series on the boundary of their maximal disk of convergence. I'm reading Bak-Newman's Complex Analysis, Chapter 18.1. A regular point ...
6
votes
0answers
46 views

Radius of convergence continuous?

Let $ f: [0,1] \rightarrow \mathbb{R} $ be analytic. Let $ r_f(x) $ be the radius of convergence of $ f $ at $ x $. Is $ r_x(f) $ continuous? Alternatively, is there an $ r_{min} $ I can choose so ...
5
votes
5answers
853 views

Polynomial $p(x) = 0$ for all $x$ implies coefficients of polynomial are zero

I am curious why the following is true. The text I am reading is "An Introduction to Numerical Analysis" by Atkinson, 2nd edition, page 133, line 4. $p(x)$ is a polynomial of the form: $$ p(x) = ...
5
votes
2answers
105 views

$\lim_{x\rightarrow\infty}\frac{f(x)}{e^x}$ for analytic functions

For some analytic function $f(x)=\lim_{n\rightarrow\infty}\sum^n_{r=0}c_rx^r$, ...
5
votes
2answers
432 views

Do all analytic and $2\pi$ periodic functions have a finite Fourier series?

Consider a function $f:\mathbb{R}\to\mathbb{R}$ which is periodic with period $2\pi$. Let us impose the condition that $f$ is analytic. Now does that imply that $f$ has a finite Fourier series? PS : ...
5
votes
1answer
87 views

A real analytic function that takes each value in $\mathbb{R}$ three times

I was inspired by this question: it is quite easy to prove that for any positive odd number $2m+1$ there exists a function $f\in C^{\infty}(\mathbb{R})$ such that ...
5
votes
4answers
306 views

Define a branch of $(z^2 − 1)^{1/2}$ which is analytic in the unit disk.

Hint: $z^2 − 1 = (z + 1)(z − 1)$. I'm really struggling with this question. I understand that for this function to be analytic it has to be differentiable in some neighbourhood, but I have no idea ...
5
votes
2answers
231 views

Using Montel's Theorem to show locally uniform convergence of analytic functions

Let $f_n :U \to \mathbb{C}$ be a sequence of analytic functions, where $U$ is open and connected. Suppose there exists a point $z_0 \in U$ such that for all $k \geq 0$ the sequence $f_n^{(k)}(z_0)$ ...
5
votes
1answer
114 views

(solution verification) the series $\sum z^{n!}$ has the unit circle as a natural boundary

I've tried to solve the following problem from Ahlfors' complex analysis text: If a function element $(f,\Omega)$ has no direct analytic continuations other than the ones obtained by restricting ...
5
votes
1answer
51 views

Analyticity of C*-algebra valued functions

Let $\mathcal{A}$ be a unital C*-algebra and consider a function $f:\mathbb{C} \rightarrow \mathcal{A}$. What is an accessible tool to prove or disprove that $f$ is analytic, i.e. can be locally ...
5
votes
1answer
83 views

Adjoint of multiplication by $z$ in the Bergman space

I am learning Hilbert space theory from Halmos' "Introduction to Hilbert space and the theory of spectral multiplicity". While talking about understanding adjoints (p. 39), he calls special ...
5
votes
1answer
145 views

How to imagine zeros of an analytic function of several variables

Let $f(z_1,\cdots, z_n)$ be a holomorphic function of several variables in an open subset of $\mathcal C^n$. Let $Z(f)=\{ (z_1,\cdots, z_n) \: | \: f=0\}$ be the zero set of $f$. If $n=1$, the zeros ...
5
votes
1answer
98 views

Proof of the three-point characterization of holomorphy

This post on Math Overflow is looking for the source of the following theorem: Let $D = \{ z \in \mathbb{C} : |z| < 1 \}$ denote the open unit disk. A function $f : D \to D$ is holomorphic iff ...
5
votes
0answers
39 views

Reference for the fact that a smooth function analytic on every line is itself analytic

Let $f \in \mathcal C^\infty(\mathbb R^p)$ ($p \geq 2$) be a smooth function such that the functions $g_d(t) := f(td)$ are all analytic for all $t \in \mathbb R$ and all $d \in \mathbb R^p.$ (i.e. $f$ ...
5
votes
0answers
79 views

Inverse of a differentiable function equal to its derivative then f is analytic

I've found a nice problem concerning analytic functions. Here it is: Let $f: (0, \infty) \rightarrow \mathbb{R}$ be a function differentiable on $(0, \infty)$ and such that $f^{-1} = f'$. Prove that ...
4
votes
1answer
122 views

If $f$ is analytic where $f$ is represented as $f=g.h$ where $g$ is analytic . From here can we conclude that $h$ is analytic?

If $f$ is analytic, where $f$ is represented as $f=g \cdot h,$ where $g$ is analytic. From here can we conclude that $h$ is analytic?
4
votes
4answers
374 views

what is the relation of smooth compact supported funtions and real analytic function?

What is the major difference between real analytic and test function (smooth compact supported functions). Can we find a real analytic function $f$ on $R^n$ which is also smooth compact supported? If ...
4
votes
3answers
589 views

The distinction between infinitely differentiable function and real analytic function

I have known that all the real analytic functions are infinitely differentiable. On the other hand, I know that there exists a function that is infinitely differentiable but not real analytic. For ...
4
votes
2answers
87 views

Number of times two rescaled, 'fully' monotonic functions can cross

Consider two functions $f: [0,1) \rightarrow \mathbb{R}$ and $g: [0,1) \rightarrow \mathbb{R}$. Suppose $f(x) > g(x)$ for all $x \in [0,1)$. Suppose further that $f$ and $g$ are infinitely ...
4
votes
2answers
301 views

Where is $\operatorname{Log}(z^2-1)$ Analytic?

$\newcommand{\Log}{\operatorname{Log}}$ The question stands as Where is the function $\Log(z^2-1)$ analytic , where $\Log$ stands for the principal complex logarithm. My understanding is that ...
4
votes
1answer
113 views

A question related to uniqueness principle theorem.

We know that the equation $ \sin^2z+ \cos^2z=1$ which holds $ \forall z \in\Bbb R$, also holds $ \forall z \in\Bbb C$. This is obvious under the shadow of following theorem: Uniqueness principle ...
4
votes
2answers
56 views

question involving analycity of $f=u+iv$

Let $f=u+iv:\mathbb C\to\mathbb C$ be analytic. Then is it true that $\dfrac{\delta^2 v}{\delta x^2}+\dfrac{\delta^2 v}{\delta y^2}=0?$
4
votes
1answer
82 views

Analytic extension for a a function defined in $\mathbb{N}$

I would like to know if it is possible to extend analytically any function of the type $f:\mathbb{N} \to \mathbb{C}$ to all complex plane. If it isn't possible, what should I assume to do so? If ...
4
votes
1answer
160 views

Entire function invariant on the coordinate axes (as sets).

From old qualifying exam: Let $E$ be the union of the two coordinate axes, i.e. $E = \{z=x+iy : xy=0\}$. Describe all entire functions satisfying $f(E) \subset E$. I feel like the best approach is to ...
4
votes
1answer
58 views

When is a Fourier series analytic?

By Fourier theory, every continuously differentiable function $f : S^1 \to \mathbf C$ admits a unique, uniformly convergent Fourier expansion $$f(\theta) = \sum_{n\in \mathbf Z} a_n e^{in\theta}.$$ ...
4
votes
1answer
165 views

Is there a $p$-adic version of Liouville theorem?

That is, if a function $f$ is analytic and bounded in all $K$, a $p$-adic field (or more generally a complete non-archimedean field), has to be constant? And does the theorem work for functions on ...
4
votes
3answers
910 views

Find a branch of $f(z)= \log(z^3-2)$ that is analytic at $z=0$.

Find a branch of $f(z)=\log(z^3-2)$ that is analytic at $z=0$. Can anyone help me on this question? I have no idea how to find a branch. The definition of branch given in lecture is $F$ is a branch ...
4
votes
3answers
433 views

Where is $\log(z+z^{-1} -2)$ analytic?

I need some help in determining where $\log(z+z^{-1} -2)$ is analytic, where $z$ is a complex number and $\log(z)=\ln|z|+\arg(z+2k\pi),k\in\mathbb{Z}$. Thank you in advanced.
4
votes
2answers
76 views

Is a function being analytic considered as a local property?

Sorry for being pedantic... I was just wondering if analyticity of a complex function considered as a local property? Apparently differentiability is considered as a local property. But analyticity ...
4
votes
1answer
330 views

Is the inverse of a real analytic function still analytic?

If $f:D\to D'$, with $D, D'$ open subsets of $\mathbb{C}$, is a complex analytic invertible function with non-zero derviative, it's easy to see that $f^{-1}:D'\to D$ is analytic too. Indeed complex ...
4
votes
2answers
179 views

Analytic off the real axis

If $f:\mathbb C \longrightarrow \mathbb C$ is continuous and $f$ is analytic off the real axis, then show that $f$ is entire.