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

learn more… | top users | synonyms

1
vote
2answers
103 views

How to solve differential equation problem involving Dirac delta function?

$$ y''+2y'+ 10y=b\,δ\left(\, t - T\,\right)\,\qquad y\left(\, 0\,\right)=3\,,\quad y'\left(\, 0\,\right)=0 $$ Can you choose values for $b$ and $T$ ( $b$ and $T$ positive numbers) such that ...
1
vote
1answer
41 views

Differential equation, for which values of 'a' does this have a bounded solution?

Let $f(t) = f(t$) be the 2pi periodic("sawtooth wave"), f(t) = t for $0 \leq t \leq 2\pi$ and consider the equation $$y^{\prime \prime} + a^2y = f$$ For which values of $a$ (here $a$ >0) does this ...
2
votes
1answer
43 views

How to find solution of the integral equation?

$$y(t) + t \int_0^t y(v)dv = 1 + \int_0^t vy(v)dv$$ I found the answer to be $y(t) = \cos{t}$. I have no idea how they go this answer. I would appreciate any suggestions how to solve this.
3
votes
1answer
48 views

A formal justification for this “physicism”?

I gave a presentation for a seminar class yesterday on Fourier analysis, and introduced the sawtooth function as a counterexample, for a function whose Fourier series is not termwise differentiable. ...
5
votes
2answers
99 views

Integrate a periodic absolute value function [duplicate]

\begin{equation} \int_{0}^t \left|\cos(t)\right|dt = \sin\left(t-\pi\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor\right)+2\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor \end{equation} I ...
1
vote
0answers
145 views

period of the sum and the product of sin(ax) and cos(bx)

Take the function f(x)=sin(ax)cos(bx) , with a,b>0 . We know that the fundamental period of sin(ax) is p= 2π/a and the fundamental period of cos(bx) is q=2π/b. Suppose there are positive integer n ...
0
votes
1answer
117 views

Period of $\sin(ax)+\cos(bx)$

Take the function $f(x)=\sin(ax)\cos(bx)$, with $a,b>0$. Suppose there are positive integer $p$ and $q$ such that $ap=bq=r$. Then $r$ is a period of $f$ but non always the shortest : for $f(x)= ...
2
votes
0answers
117 views

Is it Possible to represent $f(x) =\arctan(x)$ as a fourier series ? Why?

Is it Possible to represent $f(x) =\arctan(x)$ as a fourier series ? Why ?
1
vote
0answers
30 views

Ode with Piecewise function

We can write this $$12x"+36x'+48x=f(t)$$ my main problem is how to solve this non-homogeneous ODE I know how to do this as 2 different ode unfortunately its not in a syllabus which doesn't use ...
0
votes
1answer
74 views

Carifications on periodic functions: periodicity of $f^n, f+g, fg, \frac{f}{g}, f(g)$, constant function, etc.

During a lecture I've been given (only!) the definition of periodic function: Let $A \subset \mathbb{R}, f: A \to \mathbb{R}, t > 0$; $f$ is $t$-periodic iff for every $x \in A$, we have $a+t \in ...
2
votes
2answers
49 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$ ...
0
votes
1answer
34 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 ...
1
vote
1answer
46 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
276 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
56 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
77 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. ...
18
votes
2answers
161 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
74 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
69 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
votes
1answer
122 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
23 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
15 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
31 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
67 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
69 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
148 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
1answer
23 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
364 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 ...
5
votes
2answers
1k 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
50 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
39 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
43 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
36 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
155 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
83 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
27 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
30 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
1answer
50 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 ...
2
votes
1answer
105 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
16 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
51 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
43 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
90 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
41 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
45 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
217 views

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

Is sin(1/x) periodic? with what period time?
0
votes
2answers
55 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
72 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
236 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 $\displaystyle t_2 -t_1 = \frac{b-a}2$ ...