7
votes
1answer
97 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 ...
2
votes
0answers
38 views

Can we find a real $s$ such that $f(s)=w$ and $f'(s)≠0$?

Let $f\colon\mathbb C\to\mathbb C$ be an entire non constant function. We consider its values on the real line. The function $f$ has infinitely many real zeros and there is infinitely many real ...
1
vote
2answers
52 views

Example of a smooth 'step'-function that is constant below 0 and constant above 1

I need an infinitely smooth non-decreasing function $\ f(x)$, that $$f(x)=0\quad\forall x\leq 0,$$ $$f(x)=1\quad\forall x\geq 1,$$ and all its derivatives in $x=0$ and $x=1$ are $0$. I found that I ...
8
votes
4answers
210 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 ...
0
votes
1answer
17 views

Termwise differentiation of sequence of rational functions when the uniform limit is analytic

Given a sequence $\{f_n(x)\}$ of rational functions which converges uniformly to the analytic function $\{g(x)\}$ on $[a, b]$ ($f_n(x)$ are defined on $[a, b]$ and hence are analytic), what can we say ...
2
votes
1answer
74 views

The Identity Theorem for real analytic functions

What is the condition for two real analytic functions to be identically equal? We know that there is a nice condition (Identity Theorem) for holomorphic function to check if they are the same. What is ...
0
votes
1answer
22 views

Showing a function is not analytic

Let $g(x)=e^{-1/x^2}$ for $x\neq0$, and $g(0)=0$. I've shown that $g^{(n)}(0)=0$ and thus that it is infinitely differentiable about 0, but now I must show that it cannot be expressed as a power ...
0
votes
2answers
45 views

Proofs for $g(x)=e^{-1/x^2}$ when $x\neq0$, and $g(x)=0$ when $x=0$

Sorry for the non-descriptive title - the question is a bit long. I have $g(x)$ as in the title, and we proved previously that $g'(0)=0$ using L'Hôpital's rule. Now I must show by induction that ...
0
votes
1answer
138 views

Double exponential Taylor series $\exp(-\exp(k-ex))$

k is real constant $\gt = 1$. Is $a_n$ for $f(x)$ positive, increasing, and $\lt 1$, where $n\lt= e^{k-1}$? $$f(x) = \sum_{n=0}^{\infty} a_n x^n = \exp(-\exp(k-ex))$$ $f(x)$ is the double ...
2
votes
1answer
86 views

Smooth function becomes analytic

Let $f$ be a smooth function ,defined on unit interval $[0,1]$.Moreover $\Vert f^{(k)}\Vert_2\leq \alpha,\:\forall k\in\mathbb{N}_o$. Can we conclude that $f$ is analytic. More generally when ...
3
votes
2answers
98 views

If it converges, how to show that power series converges to $f(x)$?

I had a very basic question. Suppose $f(x)$ is a function. And let us say it has a power series :- $$f(x) = \sum_{n=0}^\infty a_nx^n.$$ Suppose we are operating inside the region of convergence. ...
4
votes
0answers
243 views

Show that the iterated $\ln^{[n]}$ of tetration(x,n) is nowhere analytic

$$f(x) = \lim_{n\to \infty} \ln^{[n]} x \uparrow\uparrow n$$ The conjecture is that $f(x)$ is monotonic and infinitely differentiable at the real axis, but nowhere analytic; because at each point on ...
4
votes
1answer
75 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 ...
21
votes
1answer
377 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 ...
0
votes
1answer
41 views

Convergence of an analytic function

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a smooth function. Let $R$ be the radius of convergence of the Taylor series centered at $a.$ For each $n \in \mathbb{N},$ let $M_n= \sup\{f^{n}(t) : t \in ...
0
votes
1answer
63 views

convolution of measurable function with analytic function

Let $f$ be a bounded measurable function with support on the unit disk $\mathbb D \subset \mathbb R^2$ and let $g$ be an analytic function on $\mathbb R^2$. Is it true that the convolution $h = f ...
18
votes
2answers
517 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 ...
0
votes
1answer
58 views

Two $C^\infty$ functions which agree on a set containing an accumulation point, but do not agree on *any* neighborhood?

As I understand it, two analytic functions defined on $\mathbb{R}^k$ which agree on a set with an accumulation point must agree on a neighborhood; however, the same is not true of $C^\infty$ ...
3
votes
2answers
154 views

Using Taylor expansion for a smooth real function

I've come across the following problem in Cracking Mathematics Subject Test, 4th Edition by Steve Leduc, from Princeton Review. Let $f(x)$ be a function that has derivatives of all orders at every ...
2
votes
1answer
172 views

Sufficient and necessary conditions to get an infinite fiber $g^{-1}(w)$

I want to verify the proof of this result and get some start ideas to overcome the different steps of this proof. Lemma: Let $g$ be a real analytic function. Then we have the equivalence ...
0
votes
1answer
37 views

Does this property persists for the derivatives $f^{(k)}, k=1,2,..$

Let $f$ be a real non polynomial analytic function. Suppose that the function $f$ assumes arbitrarily large and arbitrarily small values, i.e., for all $K>0$, there are $a,b$ with $f(a)<−K$ and ...
3
votes
1answer
75 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 ...
0
votes
1answer
85 views

What does it mean to say that a function is valued in the space of analytic functions?

I am reading some paper and I encountered this statement: ... the coefficients $a_{p,\beta}(t,x)$ [are] of class $C^m$ in $t$, valued in the space of analytic functions of $x$, in a neighborhood ...
8
votes
3answers
383 views

Smooth Pac-Man Curve?

Idle curiosity and a basic understanding of the last example here led me to this polar curve: $$r(\theta) = \exp\left(10\frac{|2\theta|-1-||2\theta|-1|}{|2\theta|}\right)\qquad\theta\in(-\pi,\pi]$$ ...
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 ...
0
votes
1answer
42 views

Can we deduce that $\lim_{x\to+\infty}f(x)=\pm\infty$ or $\lim_{x\to-\infty}f(x)=\pm\infty$?

Let $f:ℝ→ℝ$ be rael analtic function. Asume that $f$ is of finite order $1$ (An entire function is said to be of finite order if there exist numbers $a,r>0$ such that $$|f(x)|≤exp(|x|^{a})$$ for ...
0
votes
1answer
62 views

Find conditions on the function $f$ such that the fiber $f^{-1}(a)$ has a finite number of elements

Let $f:ℝ→ℝ$ be a real analytic function. We know that for any real number $a$, the fiber $f^{-1}(a)$ is a discrete set unless $f = a$. My question is: Find conditions on the function $f$ such that the ...
0
votes
1answer
123 views

Show that the fiber $f^{-1}(a)$ is finite if $a∈ℝ,a≠0$

Let $f:ℝ→ℝ$ be a real analytic function. If $f$ has infinitely many zeros, then we know that the fiber $f^{-1}(0)$ is an infinite discrete and countable set. Let $a∈ℝ,a≠0$, we know also that the fiber ...
1
vote
1answer
64 views

The fiber $f^{-1}(a)$ is a discrete (and countable) set

Let $f:ℝ→ℝ$ be a real analytic function. Then my question is: Show that for a real number $a$, the fiber $f^{-1}(a)$ is a discrete (and countable) set unless $f = a$.
0
votes
1answer
91 views

Prove there exists $f \in C^{\infty}$, bounded derivatives,

I know the title isn't very comprehensible, but I don't know how to improve it. Here is a problem which I don't know how to solve. Let $n \in \mathbb{N}, \ \ \alpha \in \mathbb{R}, \ \ \varepsilon ...
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 ...
1
vote
0answers
21 views

Analytic random function

Let $t\in[0,1]\mapsto X_t\in[0,1]$ an analytic random function. I'd like to say that the deterministic function $t\mapsto \mathbb{E}[X_t]$ is still analytic. What are the minimal conditions needed? ...
2
votes
2answers
283 views

Analytic functions of a real variable which do not extend to an open complex neighborhood

Do such functions exist? If not, is it appropriate to think of real analytic functions as "slices" of holomorphic functions?
2
votes
1answer
282 views

Show that some $C^\infty$ real function is analytic

Consider the function $f:(a,b) \rightarrow \Bbb R $, which is $C^\infty$ class. It is required to show that if the function $f$ satisfies: $\forall_{n\in\Bbb N_+} \forall_{x\in(a,b)} f^{(n)}(x) \ge 0 ...
1
vote
1answer
66 views

Continuous dependence of zeros on a parameter

Let $F:I\times J\to\mathbb{R}$ be a $C^k$ (or analytic) function, with $I,J$ real open intervals. Set $f_\lambda(x):=F(\lambda,x)$ and consider the parametric equation $$f_\lambda(x)=0\,.$$ Assume its ...
1
vote
1answer
105 views

Assume that the set of values where $f^{(k)}≠0$ is finite

Let $f:ℝ→ℝ$ be a real analytic function. Let $f^{(k)}$ be the $k$th derivative of $f$. Assume that the set of values where $f^{(k)}≠0$ is finite, then what we can say about the function $f$.
4
votes
1answer
319 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 ...
0
votes
1answer
54 views

Prove that the $k^{th}$ derivative of $f$ has necessarily infinitely many zeros

I have the following question: Let $f$ be a real entire function, i.e., $$f(x)=∑_{n=1}^{∞}a_{n}x^{n}$$ with infinitely many zeros. Prove that the $k^{th}$ derivative of $f$ has necessarily ...
1
vote
1answer
287 views

Composition Taylor Series

Is there any theorem that specifies when we are allowed to compose the taylor series of two functions? Does it have a name? Thanks.
2
votes
3answers
205 views

Entire function dominated by another entire function is a constant multiple

These two questions I didn't even find the way to solve So please if you can help Suppose $f (z)$ is entire with $|f(z)| \le |\exp(z)|$ for every $z$ I want to prove that $f(z) = k\exp(z)$ for some ...
3
votes
1answer
63 views

Notation in Old Paper

I am reading a paper http://www.ams.org/journals/bull/1935-41-04/S0002-9904-1935-06049-5/S0002-9904-1935-06049-5.pdf and I am wondering about the notation in the lemma page which states that ...
4
votes
1answer
134 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 ...
3
votes
1answer
111 views

Domain of bijectivity of function $f:\mathbb{C}\rightarrow\mathbb{C}$

There is a type of problems in my course in Complex analysis that I don't fully understand them. Given function $f:\mathbb{C}\rightarrow\mathbb{C}$, $f(z)=z^2$. You must specify the analytic and ...
1
vote
0answers
46 views

How to show that $f(x,y)$ are real analytical

Given a real function $f(x,y)$ on $R^2$, we know that (from wiki) it is called real analytic if it is locally given by a convergent power series. My question is that whether there has some principle ...
4
votes
2answers
85 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 ...
2
votes
1answer
203 views

Ring of analytic functions on the circle

Let $A = C^\omega(S^1)$ (resp. $C^\omega_{\mathbb C}(S^1)$) the ring of real-analytic real-valued (resp. complex valued) functions on the circle. These rings have maximal ideals $\mathfrak m_p = ...
14
votes
3answers
423 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 ...
5
votes
2answers
420 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 : ...
2
votes
1answer
159 views

A Paley-Wiener like theorem in real-analysis

I try to identify conditions for the Fourier-transformation $\mathcal{F}(f)$ of some function $f \in L^1(\mathbb{R}^n)$ to be real-analytic. Namely I want to show that one of the following two ...
2
votes
1answer
193 views

limit of supremum of a sequence

Can any one help me with this? Let $c$ be a real number. I would like to show that $$ \limsup_{n \to ...