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
votes
1answer
49 views

Fourier transform of $f[n]$ and $f[-n]$

Hi I am just wondering, If I have a signal $f[n]\in \mathbb{C}^L$, i.e. $f$ is $L$-periodic, i can also define $h[n]=f[-n]$. Is it true that the Fourier transform of $f$, say $\hat{F}$, and the ...
1
vote
1answer
33 views

Why is a wave with high FM aperiodic?

I was playing with sound synthesis in a program I wrote and I had a wave of the form $\sin(2\cdot\pi\cdot(f_c+\sin(2\cdot\pi\cdot f_m \cdot t)) \cdot t) $ So, just simple frequency modulation. When ...
1
vote
1answer
22 views

how can we show an antiperiodic function?

How can we graphically show an anti-periodic function? I can't imagine. Maybe I have got no imagination...!! for example we can show the sin and cos or other periodic functions on the graphs. is it ...
2
votes
0answers
24 views

Find the fundamental period

How do I find the fundamental period of this function? $$y = \sin x + \cos(1,01x)$$ I know that the fundamental period of $\sin x$ is $2\pi$ and the fundamental period of $cos(1,01x)$ is ...
3
votes
1answer
36 views

How many distinct values of floor(N/i) exists for i=1 to N.

Say we have a function $F(i)=\text{floor}(N/i)$. Then how many distinct values of $F(i)$ will exist for all $0 \leq i \leq N$ e.g. We have $N=25$ then. $F(1)=25$ $F(2)=12$ $F(3)=8$ $F(4)=6$ ...
0
votes
1answer
28 views

How to determine general solution using Laplace transform?

$y′′ + a^2y = 2u(t-10)$ Here $a > 0$ and is any real number. I am confused by $a^2$ value there. Can anyone show me step by step how to get to up to $Y(s)$? It would really help me understand. ...
19
votes
1answer
114 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 ...
1
vote
1answer
32 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 ...
1
vote
2answers
25 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 ...
2
votes
1answer
36 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
42 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
64 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
23 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
51 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
26 views
1
vote
0answers
27 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
46 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
30 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
1answer
564 views

How to prove a function is periodic?

$$f(x)=\begin{cases}1&\text{if }2n-1<x<2n,\\0&\text{if }2n<x<2n+1. \end{cases} $$ Is this function is periodic or not? How can I prove it?
2
votes
2answers
40 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
32 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
60 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
41 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
42 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
56 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. ...
3
votes
0answers
54 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 ...
7
votes
1answer
145 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 ...
8
votes
0answers
141 views

number of zeros of complex waves

Does anybody know about any type of methods how to calucalte/estimate the number of the zeros of complex waves (periodic functions as superposition of many harmonic waves) within a given period [0,x] ...
6
votes
7answers
12k 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?
0
votes
0answers
36 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
54 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 ...
0
votes
1answer
14 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
11 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 ...
1
vote
1answer
25 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, ...
2
votes
0answers
17 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 ...
1
vote
1answer
158 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 ...
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 ...
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 ...
0
votes
1answer
17 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 ...
3
votes
2answers
210 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
37 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 ...
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
39 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
29 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 ...
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 ...
1
vote
0answers
78 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 ...
0
votes
1answer
46 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 ...
0
votes
1answer
24 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. ...