Questions tagged [periodic-functions]

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

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Let $q\in\mathbb{Z} $. What is the smallest period of $\sin(2t)−\sin(qt)$?

Question: Let $q\in\mathbb{Z} $. What is the smallest period of $\sin(2t)−\sin(qt)$? My attempt: the period of $\sin(t)$ is $2\pi$, so the period of $\sin(2t)$ is $\pi$ and the period of $\sin(qt)$ is ...
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Is there any research about a function with changing "period" like sin(1/x)?

I'm encountering a function with a "changing period". It has some sense of period but not exactly. For example: f(x) = cos(1/x). Generally, it has the form of f(x + g(x)) = f(x). I cannot ...
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Time series analysis on ACF and PACF plots

So I have a non stationary time series that is hourly, daily and monthly recorded for a year and I have the ACF and PACF plots for the serie. ACF and PACF I applied the the Ljung-Box test and for ...
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Cyclic convolution of a periodic function with itself is a constant?

Let $f(x)$ be a periodic function of period $T$. Now let us define $c(x)$ the cyclic convolution (on the same period T) of $f$ with itself: $$c(x)=\int_t^{t+T}f(\tau)f(x-\tau)d\tau$$ I have the ...
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Does the Incomplete Beta function have forms of Elliptic E besides $\frac14 \text B_{\sin^2(2x)}\left(\frac12,\frac34\right)=\text E(x,2)$?

Goal: To find more special cases of the Incomplete Beta function $\text B_z(a,b)$ in terms of Elliptic $\text E(x,k)$ using Mathematica notation: The goal is to find values of: $$\text B_z(a,b)=\int z^...
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How can Floquet be used to solve periodic linear equations?

I generally understand how to solve linear systems using x' = Ax where x is a vector, and A is a matrix. However, I am lost when the A matrix becomes periodic. I am ...
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Proving that periodic vector-valued function satisfies $f(\mathbf x)= f(\mathbf x')$ iff $\mathbf x' = \mathbf x + k + \mathbf am$

In order to create a layer for a neural network that uniquely encodes relative relationships between a set of angles, I'm looking for a differentiable function with some specific properties; I'm ...
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Fourier series of a non-periodic function possible?

I have read online that Fourier series are only applicable to periodic functions. Ive included the following example in my work however it is not a periodic function, does this mean that the following ...
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What about The Catalan–Dickson conjecture for this sequence $ S(n)^{S(S(n))^{S(S(S(n))))\dots}} $ with $S(n)=\sigma(n)-n$?

It is known that aliquote sequence defined as $S(n)=\sigma(n)-n$ where $\sigma(n)$ is the sum of power divisor function (is the sum of all of $n’s$ natural divisors.),The Catalan–Dickson conjecture ...
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Given the period lengths for the orbits of $n$ different planets around the sun, how long until they all align?

Say you have $n$ planets orbiting around the sun, where the $i$th planet takes $t_i\in\mathbb{R}_{>0}$ days to complete one full cycle. Assume at $t=0$, all the planets are aligned with the sun. ...
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Continuous Periodic Functions [closed]

I was checking out the wikipedia page on periodic functions and noticed that it didn't really list very many: https://en.wikipedia.org/wiki/List_of_periodic_functions Is there a more complete ...
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Solution verification of: $f$ periodic and $\lim_{x \to \infty} f(x)$ exists implies $f$ is constant

Let $f$ be a periodic function. Show that if $\lim_{x \to \infty} f(x)$ exists in $\mathbb{R}$, then $f$ is a constant function. Use this result to prove that $\lim_{x \to \infty} \sin x$ doesn't ...
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What's the period of $\{2x\}$ [closed]

What is period of the function : $f(x)=\{2x\}$,where $\{\}$ denotes the fractional part of $x$ .
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Periodic Summation Response. How to separate its Transient and Steady-State Expression?

Background My question comes from here, it's a response of 1st order LPF RC circuit from an arbitrary periodic input. How to determine the transient response of a circuit to causal periodic inputs? ...
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How to get get period from raw data

My goal is to get suitable period from raw data. For instance, how to draw period, which is about 3, from raw data below? (Maybe, Fourier analysis can be proper tool for this, but I don't know how to ...
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Periodic Functions, LCM of periods does not exist

Can someone please give exxamples where these two statements stand incorrect? (1) Let f be non constant periodic function with periods (need not be fundamental period) T1 as well as T2 then (T1/T2) ...
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Fourier transform of an an (almost) periodic function

I am not sure about the terminology here but I have a function for which I know is almost periodic in the sense that for integer values of $k$ we have $$f(x+k)=e^{-|k|}f(x)$$ Is there any way to find ...
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Do Fourier coefficients never depend on the period T of the function?

I know many examples of periodic functions $f(t)$ where the Fourier coefficients don't depend on the period $T$. I was wondering if this is a general property or not. So far all I did was \begin{align}...
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Difference between techniques of period calculations? Continuous time vs discrete time?

For calculating period of continuous time signal,we simply divide 2pi by omega and get period value But in case of discrete time signal, procedure is not straight forward like continuous time, ...
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Any resources/bibliography for periodic solutions and periodic boundary conditions of a PDE?

can anyone recommend good books and/or articles for the mathematical analysis of periodic solutions of a PDE, for example something like $$u(t)-u''(t)=f(t) $$ with $u(0)=u(1)$ and $u'(0)=u'(1)$ ? ...
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Is $(x,y)=(0.2,0.8)$ a $4$-periodic point of the Baker's map?

The unfolded Baker's map is defined by $$ T\colon [0,1]\times [0,1]\to [0,1]\times [0,1],\qquad (x,y)\mapsto \left(2x\operatorname{ mod 1}, \frac{y+\lfloor 2x\rfloor}{2}\right) $$ Let the point $(x,y)=...
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Is it possible to impose periodic boundary conditions on Chebyshev differentiation matrix?

I have been using the Chebyshev collocation method for solving PDEs. In particular to solve a 3D generalized Poisson's equation. One of the problems I am facing is that one of my dimensions is ...
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What is the significance of modulus sign here

Let $f(x)$ be a periodic function with period $p$. Then $$f(x+p)=f(x)$$ Let, $$g(x)=f(ax+b)+c$$ Now it's mentioned in my book that $g$ is also a periodic function with a period of $\frac{p}{|a|}$. My ...
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Solution to O.D.E. is periodic.

Consider the following ODE: \begin{cases} y''(x)=-y(x)^3 \\ y(0)=1 \\ y'(0)=0 \end{cases} I have to prove that the solution is periodic. I showed that the quantity $\frac{1}{2}\left (y'(x))^2+\frac{1}{...
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Solving recursive formula dependent on a periodic function

Say we have some initial values $a_1, a_2, \dots, a_k$ and a recursive formula $a_n = f(n) + c_{1}a_{n-1} + c_{2}a_{n-2} + \dots + c_{k}a_{n-k}$ Where $f(n)$ is periodic and $a_n,f(n),c_i \in \mathbb{...
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Existence of periodic solution of a non-homogeneous ODE system

Show that the following system of ordinary differential equations $$\frac{dx}{dt}=0.5x+2.5y-x(x^2+y^2),$$ $$\frac{dy}{dt}=-0.5x+1.5y-y(x^2+y^2).$$ has at least one periodic solution. I tried to find a ...
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Eigenvalues and eigenfunctions of a discrete problem

Do you know which are the eigenvalues and eigenfunctions of the $T$-periodic discrete problem $$-\Delta_2x(k-1)=\lambda x(k), x(k)=x(k+T) (k\in\mathbb{Z}),$$ where $-\Delta_2x(k-1)=x(k+1)-2x(k)+x(k-1)$...
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Differential equation involving cross product with periodic solution

Consider a system of differential equations $$ \begin{cases} \dot{p}(t)= \frac{1}{R} p(t) \times q(t)\\ \dot{q}(t)= - p(t) \times q(t) \end{cases} \qquad (1) $$ for $p(t),q(t)\in \mathbb{R}^3$. Of ...
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Counting the number of times a periodic function crosses the $x$-axis

Suppose $f(x)$ is a smooth periodic function with period $T$ and suppose the function crosses the $x$-axis at a finite number of points $x_i \in [0,T)$, $i=1,2,\ldots,N$, within a single period $T$, ...
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What is the value set of the function $h(x) = - \frac{4}{5} \sin{\left(\frac{5\pi x}{6} \right )} - 1$

The assignment: Let us start by defining $f:\mathbb{R} \to ]-\infty,2]$ as $f(x) = - \frac{4}{5} \sin{\left (\pi x \right )} - 1$, and $g:\mathbb{R}\to\mathbb{R}$ as $g(x) = \frac{5 x}{6}$. In this ...
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Does the equality $\int _{[0,1]^n+a}f(x)d\lambda (x)=\int_{[0,1]^n}f(x)d\lambda (x)$ holds for all $a\in\mathbb{R}^n$ if $f$ is $1$-periodic?

Let $\mathfrak{B}_{\mathbb{R}^n}$ be the Borel $\sigma$-algebra of $\mathbb{R}^n$ and $\lambda :\mathfrak{B}_{\mathbb{R}^n}\to \overline{\mathbb{R}}$ be the Lebesgue measure. Definition: We say that a ...
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[Sinusoidal Function Period]: demonstrate period of $f(x)=\sin(k\,x)$

I'm an Italian student (sorry for my english). I have to demonstrate that the period $T$ of the general function $f(x)=\sin(k\, x)$ is equal to $2\,\pi/k$. I understand the idea behind the ...
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Intuition of $\left<f-a_0,f-a_0\right> =\left<f+a_0,f-a_0\right>$

Let be $\left<f,g\right>:=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}f(x)g(x)dx$, where $f$ and $g$ are $2\pi$-periodic and Riemann integrable functions, and $\frac{1}{2}a_0+\sum\limits_{k=1}^na_k\cos(...
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Finding family of functions satisfying equation

Is there a family of functions $g_{n}(t)$ (labelled by an integer index $n$ ) that satisfy the following equation?: \begin{eqnarray} \int_{0}^{T}dt g_{n}(t)e^{i(n-m)2\pi t/T}=0, \end{eqnarray} where $...
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Set of functions satisfying certain constraints

I am looking for a set of functions that satisfy some constraints. Suppose one has two periodic functions $u_{n}(\lambda,t)$ and $v_{n}(\lambda,t)$ with $\lambda, t\in\mathbb{R}$, and $n\in\mathbb{Z}$....
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Finding a periodic function with this fundamental cycle

I'd like to write a smooth ($C^{\infty}$), periodic function $f(x)$ with a fundamental cycle that looks like this: I include no scale because I'm only interested in the signs of $f(x)$, $f'(x)$, and $...
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Pendulum with Dirac Comb excitation

There is a pendulum that is excited by a Dirac Comb. $l \ddot\theta+b\dot \theta+g\theta=G\,\sum_{-\infty}^\infty\delta(t-nT)$ where $l, b, g, G$ are constants and $T=\dfrac{2\pi}{\omega}$. Show that ...
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If $f$ is periodic and $C^1$ class, then Lipschitz continuous.

I proved if $f : \mathbb R\to \mathbb R$ is periodic and $C^1$ class, then $f$ is Lipschitz continuous. I wonder if my proof is correct and if there is an easier proof. My proof is a little ...
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Lipschitz continuity of $f(x)=|x|$ on $[-1,1]$ with period $2.$

Prove the periodic function $f :\mathbb R \to \mathbb R$ s.t. $f(x)=|x|$ on $[-1,1]$ with period $2$ is Lipschitz continuous. Here is my proof. I'm stuck in the last part. Note that $f(x)=|x-2k| \ (...
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Solving for Floquet multipliers: solving over multiple periods

Everywhere that I've read about performing a Floquet analysis involves numerically solving the system over one period, with the identity matrix as initial condition. My system is fairly complex, has ...
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Can you approximate $\cos(x)$ on $\mathbb{R}$ in the space of periodic sawtooth functions?

Let any function of the form $x \mapsto 1-\frac{2x}{L}$ for $x \in (kL, (k+1)L)$ for all integers $k$ and some period $L > 0$ be referred to as periodic sawtooth. Let $\mathcal{S}$ be the space of ...
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Log periodic power Law data-fiting

Good day, I have to start working on data fitting problem, but I do not know how to start. I have found this equation can work as the curve around which data will fit. $y_t = A + B(t_c - t)^\beta[1+C \...
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Prove tha if $f:I\rightarrow\Bbb R$ is a periodic function then $\int_{x}^{x+T}f(t)\,dt=\int_0^T f(t)\,dt$ for any $x\in I$.

$$\underline{\text{**ATTENTION**}}$$ This question is not a duplicate of this one and this other question! So let be $f:\Bbb R\rightarrow\Bbb R$ a periodic function, that is there exist $T\in\Bbb R$ ...
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Does the periodic and $C^1$ class $f$ satisfy Hölder condition for all $\alpha>0$?

Let $\mathscr C^\alpha$ be a set of functions that satisfy $\alpha$- Holder condition, i.e., $$\mathscr C^\alpha:=\{f : \mathbb R\to \mathbb C \mid \exists M>0\ ;\ |f(x)-f(y)|\leqq M |x-y|^\alpha \ ...
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If $g(x)=\frac1{1-2\sin^2x}$ and $f(x)=\sin 2x$ , what is the period of $\frac{f(x)}{g(x)}$?

If $g(x)=\dfrac1{1-2\sin^2x}$ and $f(x)=\sin 2x$ , what is the period of the function $\dfrac{f(x)}{g(x)}$ ? We have $$\dfrac{f}{g}=\sin2x(1-2\sin^2x)=\sin(2x)\cos(2x)=\frac12\sin(4x)$$ Where $\cos2x\...
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How to prove this fact about peridodic distributions?

Let $f$ be a periodic piecewise continous function with period $2\pi$, and its derivative (classical derivative) is also a piecewise continous function and we denote by $\frac{df}{dx}$. Let $x_{1},...
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Q : Let $f : \mathbb Z\to \mathbb Z$ such that, for all $x,y\in\mathbb Z:$ $f(f(x) − y) = f(y) − f(f(x)).$ Show $f$ is bounded.

I came to the conclusion that f is periodic with period |f(x0)| for x0 non-zero. But I don't see how the periodicity in domain translates to a bound in the codomain. It probably has something to do ...
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Why is the period of $(\sin{\theta})^0 +(\tan{\theta})^0$, $\frac{\pi}{2}$

I found out after a few tries that the periods of both $$(\sin{\theta})^0, (\tan{\theta})^0~\text{are $\pi$}$$ Whereas i was told by my prof that period of $(\sin{\theta})^0 +(\tan{\theta})^0$ is $\...
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2 votes
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Integral Calculation or Proof of periodicity

I need help with the calculation of this integral: $$\vec H(r,a) = \int_{0}^{2\pi}\sin(\psi)\frac{\rho e^{i\psi}-re^{ia}}{(\rho^2+r^2-2\rho r\cos(\psi - a))^{3/2}} d\psi$$ Equivalently I want to know ...
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Should the domain be [0, 1] or (0, 1) for a periodic problem from 0 to 1?

I am working with a 2d periodic problem and I claim the domain to be $[0, 1]\times[0, 1]$ without much thinking. I thought since I can read a value at position 0 and 1 then it is closed. But from ...
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