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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2answers
34 views

finding the period of $\sin(2x+3)$

I tried to find the period of $\sin(2x+3)$; looking for $p>0$ such that $ \sin(2(x+p)+3)=\sin(2x+3),$ for all $ x \in R$ which means: $ \sin(2(x+p)+3) - \sin(2x+3)= 0 ,$ for all $ x \in R$ ...
1
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2answers
49 views

Periodic Functions

How to prove that a function is periodic, and find its period? Say for $\sin(2x+3)$.
0
votes
1answer
29 views

Laplace transform of a differential equation?

Find the unique solution of $y''+ y = f$, $y(0) = y'(0) = 0$ with the $2\pi$ periodic function given by $f(t)=2\pi \sin(t)$. I am having trouble setting up and starting the the question. I would be ...
0
votes
1answer
39 views

Laplace transform of $f(t)=\left|\sin\frac{t}{2}\right|$?

If you are given a rectified sine wave, $$f(t)=\left|\sin\frac{t}{2}\right|$$ how do you find the Laplace transform of this? I tried using the equation $$L\{ f(t)\} = \frac{1}{1-e^{-sT}} \int_0^T ...
0
votes
1answer
37 views

Laplace transform of a sawtooth wave

Find the Laplace transform of the periodic function such that $f(t) = t$ if $0\leq t < 2\pi$ I am having trouble setting up this question. Am I on the right path? $$ \mathcal{L}\{f(t)\} = ...
0
votes
1answer
38 views

How to determine $2\pi$ periodic function?

Let $f(t) = 2\pi \sin t$, and determine a $2\pi$-periodic function $y^∗$ with the property that $\lim_{t\to+\infty} |y(t) − y^∗(t)| = 0$ for every solution $y$ of $y′ + y = f$. I am having trouble ...
0
votes
3answers
34 views

Differentiability of a function on $\mathbb R$ such that $f(x+1)=f(x)$.

Let, $f:\mathbb R \to \mathbb R$ be a function such that $f(x+1)=f(x)$ for all $x\in \mathbb R$. Then which of the followings are correct? (a) $f$ is bounded. (b) $f$ is bounded if it is continuous. ...
4
votes
0answers
14 views

Is it possible to detect periodicity of an analytic function from its Taylor series coefficients at a point?

Given the Taylor series $\sum a_k (x - x_0)^k$ of an analytic function, it is possible to determine whether the function is periodic more-or-less directly from the coefficients $a_0, a_1, \ldots$ of ...
3
votes
0answers
28 views

Is there exact formula that returns minimal period of a periodic function?

Is there exact formula that returns minimal period of a periodic analytic function? For constant it should return 0, for non-periodic functions - infinity. I only came to the following but it ...
0
votes
0answers
30 views

Properties of periodic solutions of nonlinear ODE system

Assume you have a complicated nonlinear ODE system with some parameter $p$. Numerical simulations of the system show, that for any initial conditions and $p$, the solution tends to a periodic function ...
0
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0answers
18 views

Application for interpolating periodic B-spline

I need to draw a cubic C^2 continous, closed (periodic boundary conditions) B-spline which should interpolate a set of control points. If possible it would be great if I could specify the knot vector. ...
0
votes
1answer
12 views

Proof behind why the multiplication of two discrete time periodic sequences (with the same period) is periodic.

Given two periodic sequences $x[n]$ and $y[n]$ with the same fundamental period $N$, it is intuitive that their multiplication is also a periodic sequence with period $N$. This stems from the fact ...
0
votes
0answers
10 views

Order of distribution of the zeros of the interference function of periodic oscillations?

Given a finite (or an infinite) number of periodic oscillations of different shapes but even functions, along the abscissa $x∈R$, every such periodic oscialltion may cross on some zero points the ...
2
votes
0answers
15 views

Riemann Integrals and Periodic Functions

Consider the following result, where integrals are, say, Riemann integrals: Let $f:\mathbb{R}\to\mathbb{C}$ be a periodic function of period $L$. Then for all $a\in\mathbb{R}$ $$ ...
0
votes
1answer
43 views

If $f\in C[0,2\pi]$ and $f(0)=f(2\pi)$ then $f(\theta)=f(\theta+\pi)$ for some $\theta\,\in\,(0,\pi)$?

If $f\in C[0,2\pi]$ and $f(0)=f(2\pi)$ then $f(\theta)=f(\theta+\pi)$ for some $\theta\,\in\,(0,\pi)$? Intuitively I think it's wrong, but I failed to come up with a single counterexample. Can ...
0
votes
1answer
52 views

On the existence of a positive fundamental period

A function $f$ has period $t$ if for all $x$ in the domain it is true that $f(x+t) = f(x)$. A function is called periodic if it has (at least one) period. Take any periodic function $f$ -periodicity ...
7
votes
1answer
142 views

If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?

Let $A$ be a nonzero real number and let $B$ be a nonreal complex number. Let $z$ be a complex number. Let $f(z)$ and $g(z)$ be non-constant functions defined for all complex numbers $z$ and satisfy ...
0
votes
0answers
20 views

Extremum points of Periodic functions

Assume that the function $f,g:\mathbb{R}^n\to\mathbb R$ are $2\pi$-periodic continuous functions (so continuous functions defined on torus $\mathbb T^n$), differentiable almost everywhere in $\mathbb ...
0
votes
1answer
16 views

Determining Period From A Graph

I'm having trouble understanding what exactly the period is in a graph. My understanding is that the period is horizontal distance between 2 curves in the graph. Like in the following picture which ...
1
vote
1answer
139 views

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 ...
3
votes
2answers
202 views

How do I determine if the following function is periodic?

A question in my textbook asks me to determine if the function $f(x)=\cos(3x)+\sin(x)$ is periodic. I do not believe this to be the case as the arguments of sine and cosine are different and as such ...
0
votes
1answer
35 views

Showing a second-order ordinary differential equation has periodic orbits

Is it possible to show that $x''(t)-x'(t)²+x(t)²-x(t)=0$ has at least a periodic orbit? I've made it a system by setting $y=x'$ and get $x'=y; y'=y²-x²+x$. I'm asking the question because I find a ...
35
votes
4answers
2k views

Are there periodic functions without a smallest period?

The Wikipedia page for periodic functions states that the smallest positive period $P$ of a function is called the fundamental period of the function (if it exists). I was intrigued by the condition ...
1
vote
0answers
26 views

integral of periodic function with interval to integrate outside definition of function

So the task is to integrate $${\frac{1}T}\int_0^{2\pi}tf(t)\,dt$$ where $T=2\pi$ and $$f(t) = \left\{ \begin{array}{ll} -t^2 & \quad -\pi < t < 0, \\ t^2 ...
0
votes
1answer
37 views

Funny problem. How to average over periodic numbers

I'm want to calculate the average day and respective month that a specific event happens in a sample of countries. So for each country I have the date (one per year) of the particular event. Now my ...
1
vote
2answers
28 views

What function does this trigonometric series represent?

I wonder what known function does this trigonometric series represent? $$ f(x)=\sum_{k=1}^\infty \frac{1}{k^{3/2}}\sin{(kx)} $$ with $$ -\pi\le x\le \pi $$ and $$ f(x)=f(x+2\pi) $$ And how is it ...
1
vote
0answers
71 views

Car traveling on a bumpy road (ODE)

The suspension system of a car traveling on a bumpy road has a stiffness of $k = 5\times 10^6$ N/m and the effective mass of the car on the suspension is $m = 750$ kg. The road bumps can be considered ...
1
vote
2answers
58 views

If $f(x-1) +f(x+3) = f(x+1)+ f(x+5)$, find the period of $f(x)$.

I have an assignment full of questions like these, and I know that using some suitable substitution like replacing $x$ with $(x-1)$ and simplyifying it, I will be able to arrive at a stage where the ...
1
vote
1answer
24 views

Is the potential of a periodic conservative field periodic?

Let $Y = [0,1]^3$ and consider a conservative vector field $F$. Denote its scalar potential by $\varphi$, i.e. $$ \nabla \varphi = F. $$ If $\varphi$ is $Y$-periodic it is clear that $F$ is periodic, ...
0
votes
1answer
23 views

What function(s) satisifies $f(\theta)=-f(\theta+2 \pi )$?

This may be a trivial question, but perhaps someone can give a detailed answer. I'm looking for a periodic function that satisfies $$f(\theta)=-f(\theta+2\pi)$$ where $\theta$ is an angle in radians. ...
0
votes
0answers
14 views

Determine Periodicity with imaginary numbers

I have a function $\cos(a\cdot n) + j\sin(a\cdot n)$, for $|n| < 10$ ($a$ is some constant). My textbook says that this function is not periodic; I was wondering why this is? Is it because n is ...
0
votes
1answer
43 views

Formal solution needed to question that looks too easy to be true about the Gauss map

Using the itineraries of the Gauss map write the continued fraction expansion of the number $0 \leqslant \alpha \leqslant 1$ such that $$\displaystyle \alpha = \dfrac{1}{4+\dfrac{1}{3+\alpha}}$$ I ...
1
vote
0answers
23 views

Periodic solutions and critical points

I was going through a lecture, and for an ODE: $x' = x(5-x-2y), y'=y(-6x+x+3y)$ Which has critical points at : $(0,0) (0,2) (3,1) (5,0)$ My professor posed the question as to why the periodic ...
2
votes
1answer
94 views

Quadratic Irrationality of the Periodic points of the Gauss map

If $G:[0,1] \rightarrow [0,1]$ is the Gauss map which is defined as $$G(x) = \left\{\frac{1}{x}\right\} = \frac{1}{x} - \left\lfloor\frac{1}{x}\right\rfloor,$$ show that if $x$ is periodic of order ...
0
votes
1answer
14 views

Create a sine function to model the displacement,

When doing another experiment involving a swinging pendulum it was found that the pendulum did 48 complete swings per minute. The distance between the extreme positions of the pendulum was 6.7cm. ...
2
votes
1answer
46 views

Proving the equality of a sum and integral.

Taken from Rudin's Real and Complex Analysis text: Suppose $f$ is a continuous function on $\mathbb{R}^1$ with period $1$. Prove that $\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N f(n\alpha) ...
0
votes
0answers
33 views

Minimum number of periods to determine periodicity

Is there a minimum window window length to determine true periodicity with a given estimated period. For example, if the same event is tested for every second and is observed twice at times 1 minute ...
2
votes
1answer
70 views

If $f$ and $g$ are periodic functions, is $g \circ f$ periodic?

If $f$ and $g$ are periodic functions, is $g \circ f$ periodic? If it is, what is the period? So I know: $f(x) = f(x + T), T \in R$ $g(x) = g(x + P), P \in R$ I have this question for my homework. ...
2
votes
5answers
37 views

Proving a function $f(x + T)=k\;f(x)$ satisfies $f(x)=a^x g(x)$ for periodical $g$

I need to prove the following: If a function $\,f$ satisfies $$f(x+T)=k\;f(x), \forall x \in \mathbb R$$ for some $k \in \mathbb N$ and $T > 0$, prove that $\,f$ can be written as ...
3
votes
1answer
34 views

Detecting sinus with unknown period

I have some signal source, that can be in one of two states -- it is either emitting constant value 1.0 or oscillating in the way very close to sinus function from ...
1
vote
1answer
134 views

Is sin(1/x) periodic?with what period time? [closed]

Is sin(1/x) periodic? with what period time?
0
votes
2answers
44 views

Did I find the period of $\sqrt{1-\sin^2x}$ correctly?

$f(x)=\sqrt{1-\sin^2x} = \sqrt{\cos^2x} = \vert \cos x\vert$, period is $\pi$. Is this a correct way to find the period of this function? Can I just state that the period of $\vert\cos x\vert$ is ...
1
vote
3answers
33 views

How to find the period of the function $f(x)= 1/(2+cos(3x))$

I'm supposed to determine whether this function $$f(x)= \frac{1}{2+\cos(3x)}$$ is periodic or not, and find the period if it is periodic. According to my calculator the period is $2π/3$ but I don't ...
6
votes
2answers
219 views

determine if a function is periodic

Let $f$ be a continuous and integrable function on $[a,b]$ such that $$\int_a^b f(x)\,\mathrm{d}x = 2$$ and for every $t_1,t_2$ such that $\displaystyle t_2 -t_1 = \frac{b-a}2$ ...
1
vote
2answers
49 views

Periodic Points and One Dimensional Maps Homework Help

Let f be the tripling map $f(x)=3x\mod(1)$. I need to make a table that includes the following for $n\le6$: number of points in Fix($f^n$), number of points in Fix($f^n$) of lower period, number of ...
0
votes
2answers
72 views

How to determine the orbits of points under the tripling map $f(x)=3x\bmod 1$?

Let $f$ be the tripling map $f(x) = 3x \mod(1)$. Determine the complete orbit of the points $\frac{1}{8}$ and $\frac{1}{72}$. Indicate whether each of these points is periodic, eventually periodic, or ...
0
votes
2answers
48 views

Non-trigonometric Continuous Periodic Functions

I've seen lots of examples of periodic functions, but they all have one thing in common: They all involve at least one trigonometric term (e.g. $\sin\theta$, $\cos\theta$, etc.). My question is ...
0
votes
1answer
17 views

Period of two equal functions

I'm dealing with a problem here. We know that two functions are the same if they have the same domain and codomain. Let's say we have given the functions $f$ ang $g$ where ...
2
votes
0answers
29 views

Is a horizontal line considered periodic? [duplicate]

Given the following definition of a periodic function: $$\exists P, P > 0, f(x + P) = f(x)$$ It is possible to argue that $f(x)=k$ ($k$ being a constant) is a periodic function, since you can ...
2
votes
3answers
50 views

In what sense is a function on a circle the same as a $2 \pi$ periodic function on $\mathbb{R}$?

I was reading the appendix of Elias M Stein's Fourier Analysis and before proving the approximation lemma the author mentions the following Recall that a function on a circle is the same as a $2 ...