Articles of periodic functions

Periodic polynomial?

I was thinking if it was possible to create a polynomial that would be periodic all over the reals, since polynomials can be periodic on an interval. I then I found out the following function: $$P(x)= x \prod_{k=1}^\infty (x-k)(x+k)$$ I wonder if that function can be considered as polynomial since it’s degree will be $\infty$. […]

fundamental period

$f:\Bbb R\to \Bbb R$ is continous and nonconstant function. Let $p$ be a positive real number such that $f(x+p)=f(x) $ for all $x \in \Bbb R$ . Then there exitst $n\in\Bbb N$ such that ${p \over n }=min\{a>0|f(x+a)=f(x), \forall x \in \Bbb R\}$. Is this statement is true?

Period of derivative is the period of the original function

Let $f:I\to\mathbb R$ be a differentiable and periodic function with prime/minimum period $T$ (it is $T$-periodic) that is, $f(x+T) = f(x)$ for all $x\in I$. It is clear that $$ f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} = \lim_{h\to 0} \frac{f(x+T+h) – f(x+T)}{h} = f'(x+T), $$ but how to prove that $f’$ has the same prime/minimum period $T$? […]

Can a periodic function be represented with roots $= x^2$ where $x$ is an element of the integers?

I only want the roots of the function to equal whole squares. So the function $\sin(\pi\times x)$ would not work, the roots can only be whole square numbers.

Period of a sequence defined by its preceding term

A sequence $x_n$ is defined such that $$x_{n+1}= \frac{\sqrt3 x_n -1}{x_n + \sqrt3}, n\ge1, x_0\neq-\sqrt3 $$ We now have to find the period of this sequence. By substituting values for $x_0$ I found out the period to be $6$. Also I proved this by finding all $x_{n+k}, 1\le k \le 6$ in terms of $x_n$ […]

Minimum period of function such that $f\left(x+\frac{13}{42}\right)+f(x)=f\left(x+\frac{1}{6}\right)+f\left(x+\frac{1}{7}\right) $

Let $ f$ be a function from the set of real numbers $ \mathbb{R}$ into itself such for all $ x \in \mathbb{R},$ we have $ |f(x)| \leq 1,f(x)\neq constant $ and $$f\left(x+\dfrac{13}{42}\right)+f(x)=f\left(x+\dfrac{1}{6}\right)+f\left(x+\dfrac{1}{7}\right)$$ Prove that $ f$ is a periodic function (that is, there exists a non-zero real number $ c$ such $ f(x+c)=f(x)(c>0)$ for […]

Wave kernel for the circle $\mathbb{S}^1$ – Poisson Summation Formula

Question : How could I compute the (wave) kernel from the fact I have already found (wave) trace on unit circle? The definitions are related to the page $25$ of the following pdf. As the Spectrum$(S^1)=\{n^2 : n\ \in \mathbb{N}^*\}$, the trace (It this relevant for the question?) as distribution is simply $$w(t)=\sum_{k \geq 1} […]

Prove that $\sin(\sqrt{x})$ is not periodic, without using derivation

So basically I’m supposed to prove that identity, without using derivations. I’ve seen this question posted here already, and one answer offered derivations, another said, what I basically tried to do before coming here, to assume f(x) = f(x+T) and solve for T, but I’m not sure how to proceed on $\sin(\sqrt{x})=\sin(\sqrt{x+T})$.

Verifying if system has periodic solutions

Given the following system $\dot{x} = y$ $\dot{y} = y(9-x^2-2y^2) – x$ verify whether it has periodic solutions and if so are they attracting or repelling. I thought: The critical points or fixed point is (0,0) but is this correct and if the answer is yes then is that a periodic solution? And in general, […]

Proving a function is continuous and periodic

Suppose we are given a function $$g\left ( x \right )= \sum_{n=1}^{\infty}\frac{\sin \left ( nx \right )}{10^{n}\sin \left ( x \right )},x\neq k\pi , k\in\mathbb{Z}$$ and $$g\left ( k\pi \right )=\lim _{x\rightarrow k\pi}g\left ( x \right )$$ I found that $\lim _{x\rightarrow k\pi}g\left ( x \right )= \sum_{n=1}^{\infty}\frac{1}{10^{n}}$ for both odd and even $k$. However, […]