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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1answer
250 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
478 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
411 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
343 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
318 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
307 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
122 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
3answers
309 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
2answers
451 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: ...
6
votes
3answers
859 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
341 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
323 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
496 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
183 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
322 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 ...
5
votes
5answers
761 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
4answers
777 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 ...
5
votes
4answers
227 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
178 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
2answers
365 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
48 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
65 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
89 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
92 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
73 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
120 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
324 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
476 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
84 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
1answer
87 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 ...
4
votes
2answers
240 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
83 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
55 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
153 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
158 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
820 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
2answers
73 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
271 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
162 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.
4
votes
1answer
130 views

Mean values theorem and countable sets

The mean values theorem says that there exists a $c∈(u,v)$ such that $$f(v)-f(u)=f′(c)(v-u)$$ My question is: Assume that $u$ is a root of $f$, hence we obtain $$f(v)=f′(c)(v-u)$$ Assume that $f$ is a ...
4
votes
1answer
391 views

Analytic functions and Fourier Series

I'm taking my first real analysis course and I'm trying to get a better feel about analytic functions. My understanding is that an analytic function is one which can be written as a power series. My ...
4
votes
1answer
50 views

Commutativity of integration and Taylor expansion of the integrand in an integral

I am baffled with a seemingly a straightforward problem. Suppose we are given the following integral: \begin{equation} f(a)\,=\,\int_{0}^{\infty} \frac{x^4}{x^4+a^4} e^{-x}, \end{equation} and we ...
4
votes
1answer
202 views

composition of non analytic functions can be analytic?

I was wondering if the following scenarios are possible: 1) composition of an analytic function with a non-analytic continuous function being analytic (except for trivial cases) 2) composition of two ...
4
votes
0answers
424 views

Convergent sum with primes

If $f(n)$ is a strictly increasing elementary function from the reals to the reals, and $p(n)$ is the $n$'th prime number. Is there any $f(n)$ such that $\sum_{n=1}^\infty\frac{1}{f(p(n))}$ is ...
3
votes
4answers
269 views

If $f$ is holomorphic and $\,f'' = f$, then $f(z) = A \cosh z + B \sinh z$

Suppose $f$ is holomorphic in a disk centered at the origin and $f$ satisfies the differential equation $$f'' = f.$$ Show that $f$ is of the form $$f(z)=A \sinh z + B \cosh z,$$ for suitable constants ...
3
votes
3answers
215 views

$|f|$ constant implies $f$ constant?

If $f$ is an analytic function on a domain $D$ and $|f|=C$ is constant on $D$ why does this imply that $f$ is constant on $D$? Why is the codomain of $f$ not the circle of radius $\sqrt{C}$?
3
votes
1answer
58 views

Largest domain on which $z^{i}$ is analytic.

Can anyone help me with this question: What is the largest domain $D$ on which the function $f(z)=z^{i}$ is analytic?
3
votes
1answer
149 views

Basic question about analyticity vs. differentiability in complex analysis.

In chapter $V$ of Palka, "Consequences of the Local Cauchy Integral Formula," 3.1. If a function $f$ is analytic in an open set $U$, then $f'$ is analytic in $U$. In particular, $f$ belongs to ...
3
votes
1answer
72 views

Analyticity of a real function on $[0,\infty)$

I'm struggling to understand the difference of the analyticity of a real and a complex functions. Consider the following real valued function which is a minimal example of a somewhat more involved ...