Questions on periodic functions, functions $f(x)$ that satisfy the identity $f(x+c)=f(x)$, for some nonzero $c$.

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2
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3answers
480 views

How to find the period of a periodic function?

I'm working on some numerical analysis problem, and I'm studying functions that "seem" to be periodic. Now what I would like to do, is to determine their period. Only, the methods I actually use are ...
0
votes
1answer
71 views

how can I tell the difference bewteen chaos and periodic with lots of noise?

I am most interested in the difference of power spectrums between chaos and periodicity. I read some paper on testing chaos against some more general alternatives (mostly stochastic systems) through ...
2
votes
0answers
49 views

Do there exist periodic fractals $A_f$ of this type?

Let $z$ be a complex number. Meromorphic here means meromorphic on all of the complex plane $C$. Lets define a fractal $A_f$ on the complex plane as the result of iterating a meromorphic function ...
3
votes
2answers
184 views

Show that a the periodic function is even in a specific interval

I have just started to learn about Fourier series, and Even/Odd functions. I am supposed to show that the function below is even in the given period. I assumed that if I tried solving the $B_n$ it ...
-1
votes
1answer
77 views

Prove the orbits of ODE system to be periodic?

For the parallel flow $\dot{\theta_1}=w_1$, $\dot{\theta_2}=w_2$ on the 2-torus, where $w_1$ and $w_2$ are positive and where the coordinates $\theta_1$ and $\theta_2$ are taken modulo 1. Also, ...
6
votes
2answers
707 views

Is $f(x)=\sin(x^2)$ periodic?

Is the function $f:\Bbb R \rightarrow \Bbb R$ defined as $f(x)=\sin(x^2)$, for all $x\in\Bbb R$, periodic? Here's my attempt to solve this: Let's assume that it is periodic. For a function to ...
1
vote
2answers
145 views

Asymptotic Expansion of a Multiscale Partial Differential Equation

I'm trying to understand how to solve $$-\nabla\cdot(K(\frac{x}{\epsilon})\nabla u(x,\frac{x}{\epsilon})=f \text{ in } \Omega$$ $$u(x,\frac{x}{\epsilon})=u_D \text{ in } \partial \Omega$$ where ...
1
vote
0answers
90 views

Inequalities of integrals of periodic functions

I have a function that has a shape similar to $\sin(x)^2$ (could be periodic extensions of $(x/(\pi/2))^2$ defined between $-\pi/2$ to $\pi/2$ for example). Let's call it $g(x)$. I want to show that ...
6
votes
5answers
7k views

How to find period of this periodic function?

How can I find a period of this function? $$2\sin{3x} + 3\sin{2x}$$ Is here any way how to sum both sinuses?
3
votes
3answers
159 views

Does anyone know of any additive periodic functions?

Anyone know of any periodic functions satisfying $f(xy)=f(x)+f(y)$, when gcd(x,y)=1, I need a function other then the function $a_d(k)$, thats 1 if d divides k, and 0 if it doesn't.
2
votes
2answers
102 views

calculate the period of an hypotrochoid

I'm curious how to find out the period of an hypotrochoid. x = (a-b) * cos(t) + h * cos( ((a-b)/b) * t ) y = (a-b) * sin(t) - h * sin( ((a-b)/b) * t ) I know ...
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votes
0answers
63 views

Factoring from order finding?

It appears that if you have a finite field, F_p, and you can factor the order of the field, p-1, then you can easily construct an arbitrary nth root of unity in F_p and also check that it be primitive ...
6
votes
3answers
208 views

prove a function to be constant

Let the function $f\colon \mathbb{R} \to\mathbb{R}$ have arbitrarily small positive periods, in the sense that if $\delta>0$ then there exists $T \in (0,\delta)$ such that $f(t + T) = f(t)$ for all ...
1
vote
1answer
43 views

How to prove the periodicity of an iterated function?

How to prove the periodicity of an iterated function? For example, how to prove $\sin_{[n]}(x)$ is a periodic function of period $2\pi$ $\forall n\in\mathbb{R}^+\cap\{0\}$ ?
0
votes
1answer
74 views

For the sum to be periodic, $f_1$ and $f_2$ must be commensurable;

For the sum to be periodic, $f_1$ and $f_2$ must be commensurable; that is, there most be a number $f_0$ contained in each an integral number of times. Thus, if $f_0$ is the largest such number, ...
2
votes
0answers
234 views

Fourier Series problem

Suppose you are given the following information about a continuous-time periodic signal, $x(t)$, with period $6$ and its Fourier series coefficients $(a_k)$, (1)-(4). Using the synthesis equation, ...
2
votes
2answers
572 views

Finding additional function values of an odd-periodic function.

I'm in a calc I class where I'm faced with the question: Suppose that f(x) is an odd function, and periodic with period 10. If f(3) = 4, find f(7) + f(5). Unfortunately, this is not talked about ...
0
votes
3answers
98 views

Is $y(t) = t^2 + i\cdot t^2$ a periodic function?

Is complex valued function like $y(t) = t^2 + i\cdot t^2$ a periodic function?
8
votes
2answers
154 views

Are there functions that have $\Re(f(z))$ periodic but $f(z)$ is not periodic?

Let $f(z)$ be a function meromorphic in a simply connected convex domain $D$ (subset of the complex plane with positive area or the whole complex plane) where $z$ is a complex number. Are there such ...
5
votes
1answer
91 views

Functions whose $n^{th}$ Derivatives form a cycle

The simplest example of this is $e^x$ which we could say has period 1 (it is its own derivative). $e^{-x}$ would have period 2. Using similar constructions, I can get a function that has a derivative ...
3
votes
0answers
458 views

Integration of nontrivial trigonometric functions

First an example which I know how to solve. If we have the following integral $$\int_{-\pi}^{\pi}\frac{1}{1+3~\cos^2(t)}dt$$ there is a very practical way to evaluate it by interpreting it as some ...
2
votes
1answer
124 views

Relation on fourier coefficients implies smoothness for a periodic continuous function

I just came across with the following question.. suppose we are given a periodic function of period $2\pi$. We define $a_n$ and $b_n$ to be the Fourier coefficients of $f$. To be precise, we have ...
8
votes
3answers
173 views

Integral inequality on a periodic function

Given $f:\mathbb R\to \mathbb R^+$ continuous and periodic of period $T\geq 0$, I am asked to prove that $$\int_0^T\frac{f(x)}{f(x+\alpha)}\mathrm dx\geq T,\; \forall \alpha\in\mathbb R.$$ How does ...
5
votes
1answer
294 views

Elliptic functions and (meromorphic) simply periodic functions.

Let be $f:\mathbb C\rightarrow\overline {\mathbb C}$ a meromorphic function. The set of periods $\Omega_f$ is a discrete (additive) group and we have one of these possibilities: i) $\Omega_f= \{0\}$ ...
2
votes
2answers
723 views

Periodic solution of differential equation

let be the ODE $ -y''(x)+f(x)y(x)=0 $ if the function $ f(x+T)=f(x) $ is PERIODIC does it mean that the ODE has only periodic solutions ? if all the solutions are periodic , then can all be ...
1
vote
2answers
1k views

Proof that a periodic function is bounded and uniformly continuous.

I need to show that if $f:\mathbb{R}\to \mathbb{R}$ is continuous and $\forall x \in \mathbb R, f(x+1)=f(x)$, then: $f$ is bounded, $f$ is uniformly continuous, there exists $c\in \mathbb{R}$ such ...
1
vote
3answers
626 views

When is the integral of a periodic function periodic?

I'm attempting some questions from Zwiebach - A First Course in String Theory, and have got stuck. I've proved that a function $h'(u)$ is periodic. The question then asks me to show that ...
1
vote
1answer
139 views

How can you find the 3d period of a summation of plane waves?

I realize this is a very hard question. In the very least I'd like to know if there is a way to do this or not. Say you have a summation of plane waves in a 3d volume, with longitudinal and ...
0
votes
2answers
345 views

Period of a finite binary sequence

Let $G:N\to\{0,1\}$, and let $L$ be some period of $G$, so that $G(i+kL)=G(i)$. What's the best a good way to find the smallest period of $G$? I mean an algorithm that takes ($G$,$L$) and outputs the ...
4
votes
2answers
449 views

Is the continued fraction of the square root of a base $\phi$ (golden ratio) number periodic when the continued fraction is expressed in base $\phi$?

I have been looking at concise ways to represent irrational numbers using only integers. I was thinking about base $\phi$ (golden ratio base) and how it can represent the quadratic extension of the ...
1
vote
0answers
105 views

sum of periodic function which eventually vanish

It is known that : If $\lim_{x \to \infty} \sum_{i=1}^n f_i(x)=0$ and $f_i$ are real-valued periodic function, then $\sum_{i=1}^n f_i(x)=0$ for all $x \in \mathbb{R}$. We can solve this problem by ...
3
votes
1answer
103 views

Antiperiodic functions are indecomposable

Let $A$ be the algebra of continuous functions $\mathbb{R} \to \mathbb{R}$ which are periodic of period $1$, and write $M$ for the $A$-module of antiperiodic functions of period $1$ (meaning $f \in M$ ...
0
votes
1answer
121 views

How can a Bézier curve be periodic?

As I know it, a periodic function is a function that repeats its values in regular intervals or period. However Bézier curves can also be periodic which means closed as opposed to non-periodic which ...
5
votes
1answer
7k views

Period of the sum/product of two functions

Suppose that period of $f(x)=T$ and period of $g(x)=S$, I am interested what is a period of $f(x) g(x)$? period of $f(x)+g(x)$? What I have tried is to search in internet, and found following link ...
0
votes
1answer
293 views

Find the period of the following function

suppose that we have function $y=[2x]-3*[4x]$ here $[*]$ denotes as a minimum distance till integer. we are required to find period of this function,first of all i am confused in terms of ...
1
vote
2answers
79 views

a Function with several periods

A periodic function is given by $ f(x+nT)=f(x) $, with 'n' an integer and T the period. My question is if we can define a non-constant function with several periods; by that, I mean $ ...
1
vote
0answers
32 views

Asymptotic order of some sums with the Fourier coefficients

Given $f\in C^{w}[0,1]$ with periodic conditions $f(0)^{(j)}=f(1)^{(j)},\ j=0,\dots, w-1$ and its Fourier series are $f(x)=\sum_{l}f_{i}\exp(2\pi ix)$. I need to find the asymptotic order of errors ...
0
votes
1answer
232 views

Fourier Coefficients of periodic function

Consider a Function $f\in L^2(\mathbb{T})$. Is there any lower bound for the decay of the Fourier coefficients $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt$$ known? There are a lot ...
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0answers
103 views

Non-periodic BV function

I want to know the definition of non-periodic bounded variation function. I know the definition for periodic function of bounded variation, which is, Let $f:[a,b]\to \mathcal c$ and ...
2
votes
1answer
85 views

To show $\int_{-\pi+x}^{\pi+x}f(u) du=\int_{-\pi}^{\pi}f(u) du$. [duplicate]

Possible Duplicate: An integrable and periodic function $f(x)$ satisfies $\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)dx$. I have to show if $f\in \mathcal L[-\pi,\pi]$ and if $$f(u+2\pi)=f(u), ...
2
votes
4answers
96 views

Variation of periodic function

If $f\colon\Bbb R\to\Bbb R$ then $\operatorname{var}(f, [a,b]):=\sup \{\sum_{k=1}^n |f(x_k)-f(x_{k-1})| \}$, where supremum is taken over all finite sequences $(x_k)$ such that ...
1
vote
1answer
228 views

How to effectively compute a periodic function?

I'm writing a program to compute a value of periodic function for any arbitrary large argument: $f(k) = (\sum_{n=1}^{2^k} n)\mod\ (10^9 + 7)$, where $n,k \in \mathbb{N} $ I know that $ f(k + P) = ...
3
votes
1answer
203 views

How to construct and oscillation with exponentially growing period times?

I'm searching for the (maybe even smooth) "oscillating" function $$f(t)=A\sin{\left(g(t)\right)},$$ such that there are zeroes at times $t_n=T^n$ for some fixed number $T$. So this will not really ...
6
votes
3answers
946 views

What is the periodicity of the function $\sin(ax) \cos(bx)$ where $a$ and $b$ are rationals?

So, I have a general question first. What happens to the periodicity when we multiply two periodic trig functions with one another ? The next one is very specific, what is the period of the function ...
2
votes
1answer
151 views

How to determine periodicity of complex log in different bases?

How do you determine the "period" of a complex logarithm as a multivalued function in an arbitrary (real or complex) base? I apologize in advance if my terminology is incorrect, but let me illustrate ...
2
votes
2answers
268 views

Limit of integral involving periodic function

Can anyone help me with this? I want to know how to solve it. Let $f:\mathbb R \longrightarrow \mathbb R$ be a continuous function with period $P$. Also suppose that ...
1
vote
1answer
365 views

What's the prime period of $\frac{\sin 2x + \cos 2x}{\sin 2x - \cos 2x}$

What's the prime period of the following function? $$\frac{\sin 2x + \cos 2x}{\sin 2x - \cos 2x}$$
5
votes
1answer
137 views

Triangular periodic tessellation in two variables

The simple function $f(x, y) = \cos(x) + \cos(y)$ is doubly periodic in a square tessellation. One feature of this surface is that cuts through the nearest extrema in either of two directions (e.g. ...
0
votes
2answers
37 views

Is it okay to use the same variable to describe function periods?

If I have a system of period functions of $x$ and $y$, in this case trigonometric, $$\begin{cases} \sin{(2x + y)} = 0 \\ \sin{(2y + x)} = 0 \end{cases}$$ is it okay for me to to use the ...
2
votes
1answer
564 views

FFT for solving Poisson equation in 3 dimensions with periodic boundary conditions

In electrostatics, the laplacian of the electrostatic potential $\Delta V(\mathbf{r})$ created by a charge distribution $\rho(\mathbf{r})$ is $$ \Delta ...