All questions with tag [math: analyticity]


Can a function "grow too fast" to be real analytic?

Does there exist a continual function $\: f : \mathbf{R} \to \mathbf{R} \:$ such that for ¢ all real analytic features $\: g : \mathbf{R} \to \mathbf{R} \:$, for all actual numbers $x$, ¢ there exists an actual number $y$ such that $\: x < y \:$ and also $\: g(y) < f(y) \:$?
2022-07-25 07:47:36

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 : Although this question seems to be related to this, I couldn't find anything that I can understand there
2022-07-25 07:39:28

Calculating germs for complete holomorphic function

I'm trying to find the germs $[z,f]$ for the complete holomorphic function $\sqrt{1+\sqrt{z}}$ . The question indicates that I should find 2 germs for z = 1, but I seem to be able to find 3! Where have I gone wrong? I start by defining $U_k={\frac{k\pi}{2}&lt\theta&lt\pi+\frac{k\pi}{2}}$, then ${(U_k, f_k)}$ for $0\leq k \leq 15$ is an e...
2022-07-22 15:33:18

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 conditions is sufficient: $\exists K \text{ compact}: \text{supp}(f) \subset K$ $\exists C\in (0,\infty): |f|\leq\exp(-C|x|)$ A few notes: Certainly, w...
2022-07-22 15:12:41

Analytic non constant function

Stuck up on something in complex analysis. Let $f$ analytic function and open $\Omega \subset \mathbb{C}$. Show that if $f$ is not a constant on a neighbourhood of $z_0$, then exist a neighbourhood $V$ of $z_0$ so that $z\in \mathbb{V}$ and $f(z)=f(z_0) \Rightarrow z=z_0$. Note: This should be proven without Cauchy-Riemann because of the axioma...
2022-07-22 12:21:44

If $f$ is a non-constant analytic function on a compact domain $D$, then $Re(f)$ and $Im(f)$ assume their max and min on the boundary of $D$.

This is a homework problem I got, my attempted proof is: Since $f$ is non constant and analytic, $f=u(x)+iv(y)$ where neither $u$ nor $v$ is constant(by Cauchy Riemann equations) and $u v$ are both analytic in $D$. Therefore $u$ and $v$ both assume their max on the boundary of $D$ (by maximum modulus theorem). Also, $u$ and $v$ have no minimu...
2022-07-11 20:46:14

Area and locally one-to-one analytic mappings of the unit disk.

We learned about conformal mappings and various properties of locally one-to-one, analytic mappings of the unit disk. I am having trouble with the following problem, can anyone help? Let $f(z) =\sum_{n=0}^\infty a_n z^n$ be analytic and locally one-to-one in the unit disk, $|z| \leq 1$, and suppose $f$ maps the unit disk onto a domain $D$ whose...
2022-07-09 20:16:05

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

I have the adhering to trouble. Allow $f:D\to \mathbb C$ be a holomorphic function, where $D=\{z:|z|\leq 1\}.$ Let $$f(z)=\sum_{n=0}^\infty c_nz^n.$$ Let $l_2(A)$ represent the Lebesgue action of a set $A\subseteq \mathbb C$ and also $G=f(D).$ Prove that $$l_2(G)=\pi\sum_{n=1}^\infty n|c_n|^2.$$ After a lengthy battle I took care of ahead up...
2022-07-06 00:39:21

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

From old certifying test: Let $E$ be the union of both coordinate axes, i.e, $E = \{z=x+iy : xy=0\}$ Describe all whole features pleasing $f(E) \subset E$. I seem like the most effective strategy is to take into consideration the power collection of $f$. My first strategy was to list restraints by taking into consideration the function related ...
2022-07-05 21:35:21

limit of supremum of a sequence

Can any kind of one aid me with this? Allow $c$ be an actual number. I would love to show that $$ \limsup_{n \to \infty}\sqrt[n]{\left|\frac{i}{2}\left(\frac{(c-i)^{n+1}-(c+i)^{n+1}}{c^{2}+1}\right)\right|}=\sqrt{c^{2}+1}.$$ I thought of this when I was attempting to confirm that the distance of merging for the power series of after that funct...
2022-07-04 17:52:31

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

That is, if a function $f$ is analytic and also bounded in all $K$, a $p$ - adic area (or even more usually a full non - archimedean area), needs to be constant? And also does the theory benefit features on $K^n$, or in $\mathbb{C}^n$?
2022-07-04 14:56:28

proving the function $\frac{1}{1+x^2}$ is analytic

I have an inquiry, just how can we confirm that a function, below specificaly the function $\frac{1}{1+x^2}$ is analytic? I recognize we must show that for any kind of $x_0$ in $\mathbb R$, the collection $\sum_{k=0}^{\infty}\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k$ has a convergent distance more than absolutely no, yet just how to show that? I value yo...
2022-07-04 14:15:51

Analytic functions and Fourier Series

I'm taking my first real analysis training course and also I'm attempting to get a far better feeling concerning analytic features. My understanding is that an analytic function is one which can be created as a power series. My understanding is that a power series is just one of the kind $\sum_n a_nx^n$. I was reflecting to Fourier series and al...
2022-07-03 01:09:06

A trivial fact about analytic functions

In guide that I'm reviewing they claimed that this reality is unimportant, yet I'm not entirely certain concerning something and also I favor to validate it. It has to do with a great deal of equivalences, yet I have inquiry concerning 2. Allow $f$ be a function specified on a domain name $D \subset {\Bbb C}$. Think that $f$ has a power collect...
2022-06-29 22:31:32

Convergent sum with primes

Let $f(n)$ be a purely raising elementary function from the reals to the reals, and also allow $p(n)$ represent the $n^{\rm th}$ prime number. Exists any kind of $f(n)$ such that $\sum_{n=1}^\infty\frac{1}{f(p(n))}$ is algebraic, or has a closed form in regards to primary features? Otherwise, exists a method to confirm that the amount must be t...
2022-06-27 17:28:17

Extending to a holomorphic function

Let $Z\subseteq \mathbb{C}\setminus \overline{\mathbb{D}}$ be countable and also distinct (below $\mathbb{D}$ represents the device disc). Take into consideration a function $f\colon \mathbb{D}\cup Z\to \mathbb{C}$ such that 1) $f\upharpoonright \overline{\mathbb{D}}$ is continual 2) $f\upharpoonright \mathbb{D}$ is holomorphic 3) if $|z_0|=...
2022-06-25 10:01:39

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

Hilbert is 19th trouble asks: Are the remedies of normal troubles in the calculus of variations constantly necessarily analytic? This was confirmed to be real (via the job of Sergei Bernstein, Ennio de Giorgi, John Nash, to name a few). My inquiry possibly stems primarily from my primary expertise of the topic, yet I am questioning just wha...
2022-06-25 08:41:20

Analytic continuation of Fourier transform

Let $$h(u)= \int_{-\infty}^{\infty} g(x) \exp(iux) \, \text{d}x$$ be the Fourier change. After that allow us intend that for $ |x| \to \infty $ the function $g$ goes as $$g(x) = \exp(-ax) \text{ for some positive $a$}$$ Does this mean that we can analytically proceed the Fourier change $h$ to the area of the facility aircraft where $-a &...
2022-06-24 23:27:52

Liouville's theorem problem

Hello there i require some tips and also aid with this trouble. Allow $f\in\mathcal O(\mathbb C)$ and also think that $\Re f(z)\geq M$ for all $z\in\mathbb C$. Usage Liouville´s theory to confirm that $f$ is constant function. I am actually stuck on this trouble.
2022-06-10 17:37:41

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

I wonder why the following holds true. The message I am analysis is "An Introduction to Numerical Analysis" by Atkinson, 2nd version, web page 133, line 4. $p(x)$ is a polynomial of the kind: $$ p(x) = b_0 + b_1 x + \cdots + b_n x^n$$ If $p(x) = 0$ for all $x$, after that $b_i = 0$ for $i=0,1,\ldots,n$. Why is this real? As an exam...
2022-06-09 16:14:07