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

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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
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3answers
215 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 ...
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1answer
44 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 $2\pi$ $\forall n\in\mathbb{R}^+\cap\{0\}$ ?
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1answer
81 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, ...
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0answers
245 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, ...
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2answers
638 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 ...
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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?
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2answers
159 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 ...
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1answer
98 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 ...
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0answers
497 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 ...
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1answer
133 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 ...
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3answers
187 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 ...
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1answer
341 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
790 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 ...
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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 ...
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3answers
727 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 ...
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1answer
149 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 ...
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2answers
398 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 ...
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2answers
493 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 ...
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0answers
106 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 ...
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1answer
109 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 $f \in M$ ...
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1answer
140 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 ...
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1answer
8k 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 ...
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1answer
335 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 ...
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2answers
83 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 $ ...
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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 ...
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1answer
243 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
109 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
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1answer
88 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), ...
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4answers
99 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 ...
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1answer
243 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
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1answer
215 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 ...
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3answers
1k 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
158 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
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2answers
283 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 ...
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1answer
395 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
139 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. ...
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2answers
38 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
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1answer
674 views

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

In electrostatics, the laplacian of the electrostatic potential $\Delta V(\mathbf{r})$ arising from a charge distribution $\rho(\mathbf{r})$ is $$ \Delta ...
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3answers
1k views

Does the phrase “instantaneous frequency” make sense?

I had always thought of time and frequency as being two different (yet complete) descriptions of the same system, so to me, the phrase "instantaneous frequency" didn't make sense -- frequency is a ...
0
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1answer
109 views

Function with oscillating frequency?

I'm looking for a function whose frequency oscillates around a certain value (say, oscillating between 440 Hz and 880 Hz, at a rate of 1 Hz -- i.e., its frequency goes up and down once per second, ...
4
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4answers
1k views

Are sin and cos the only continuous and infinitely differentiable periodic functions we have?

Sin and cos are everywhere continuous and infinitely differentiable. Those are nice properties to have. They come from the unit circle. It seems there's no other periodic function that is also ...