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

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3
votes
2answers
48 views

Period of $f(2x+3)+f(2x+7)=2$

I have a problem in finding the period of functions given in the form of functional equations. Q. If $f(x)$ is periodic with period $t$ such that $f(2x+3)+f(2x+7)=2$. Find $t$. ($x\in \mathbb R$) ...
0
votes
1answer
15 views

Complex Eigenvalue Periodic

This is probably a simple question, but I'm having trouble finding a clear answer. Let's say we have a system, with the complex eigenvalue: $$\lambda = \alpha + i\beta$$ I know that $\alpha < 0$ ...
4
votes
2answers
82 views

detect largest period in non-harmonic components

let us consider following sinusoidal components $\sin(2\pi 13.5t)+\sin(2\pi 13.99t)+\sin(2\pi 25.3t)+\sin(2\pi 26t)$, clearly this is not periodic in total,because frequencies or periods are not ...
0
votes
2answers
16 views

Condition of periodic function for |sin πx|

Period of |sin πx| = 1 Wolfram alpha : So why this condition for Periodic function is not true? f(x) = f(x + T) Wolfram alpha :
4
votes
2answers
339 views

How to show that this real function is not periodic?

How can one prove that $$\cos\left(\frac{\pi}{2} t \right)+\cos\left(t \right)$$ is not periodic? This question is motivated by the harmonic spectral representation of time series. Indeed, it is ...
2
votes
2answers
105 views

I have a differential equation which solution is periodic. What can I tell about right-hand side of such equation?

I have equation of form $$ \frac{dx}{dt} = f(x), $$ and know and for some initial value $x_0$ its solution is periodic with unknown period. What can I tell about $f(x)$ apart from non-linearity (or ...
0
votes
1answer
272 views

Laplace transform of a periodic function

Knowing that $$L[f(t)]=\frac1{1-e^{-sp}}\int_0^{p} e^{-st}f(t)dt$$ $p$ indicates the period of the function If $f$ is a continuous function by segments in $[0,\infty)$ and $F(s)=L[f(t)]$ exists for ...
0
votes
0answers
21 views

How to find p such that 1/p has a repeating decimal with a specified period

Alright, so I know a bit of information about the problem but I'm having trouble tying it all together. I know: if gcd(n,10)=1 then 1/n has a repeating decimal expansion. 10^3 = 1 mod p 1/p = ...
2
votes
1answer
334 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?
0
votes
1answer
39 views

Periodic absolute value function

Define $$h(x)=|x|$$ on the interval $[-1,1]$ and extend the defintion of $h$ to all of $\mathbb{R}$ by requiring that $h(x+2)=h(x)$. Now define the function: $$h_n (x)=\frac{1}{2^n} h(2^n x)$$ ...
0
votes
1answer
35 views

Addition of two cosine waves with different periods

I was just wondering if anyone knows how to add two different cosine equations together with different periods to form one equation. Is there a way to do this and get a real answer or is it just all ...
0
votes
0answers
19 views

Linear Differential Equation with Quasiperiodic Coefficient: A Question

I am reading the book Synergetics. The following passage is from the book: Assume that in the differential equation $\dot{q}= a(t)q$, where $a(t)$ is quasiperiodic, i. e., we assume that $a(t)$ ...
1
vote
0answers
19 views

Analyzing a recurrence model: equilibriums, stability and periodic behavior.

In orer to increase my knowledge in math I decided to analyze the following recurrence relation (logistic growth in ecology) $$N(t+1) = N(t) (1 + r(1-\frac{N(t)}{K}))$$ I found the equilibriums by ...
-1
votes
1answer
29 views

Creating a formula from data

$(2711, 0.62),(3243,1.83),(3846,0.38),(4514,2.42),(5152,0.58),(5723,1.82),(6322,0.38), (6950, 2.44),(7628, 0.57),(8159,1.82),(8757,0.39),(9425,2.44),(10102, 0.56), (10635, 1.82),(11230, 0.41),(11858, ...
0
votes
0answers
9 views

analytic property of periodic properties

Can any one suggest me some books in which I can see the analysis of periodic functions? I dont have any constrain in the domain or codomain. For example this book ...
1
vote
0answers
40 views

Definite Integral of Periodic Function Multiplied by another Function

For one part of a problem I am working on, I am trying to show that $y'(t) \geq 1$ for all $t \geq 0$ when $y'= 1- \int^t_0 g(s)y(s) ds$ When $g(t)$ is periodic, $g(t) <0$ for all $t$, and ...
1
vote
2answers
62 views

Periodic sequence [duplicate]

$(x_n)_n$ is a sequence defined by the relation: $x_{n+2}=|x_{n+1}-x_{n-1}|$ for $n\geq1$ and $x_0,x_1,x_2$ are non-negative integers, not all three equal 0. I think this sequence is periodic, so here ...
6
votes
5answers
193 views

Sum of two periodic functions is periodic?

I have following paragraph taken from the Stanford's study material. Question: Is the sum of two periodic functions periodic? Answer: I guess the answer is no if you are Mathematician, yes ...
0
votes
0answers
13 views

Mathematical project on ocean waves and buoyancy

I am currently searching for an idea for a math project on actual ocean waves and water volume displacement. I am thinking of creating a general formula or ratio of crest to trough volume for when a ...
2
votes
2answers
271 views

Extend a function as odd/even periodic function

Let $f$ be the function $f(x) = x^2 + 2 $, where $ 0<x<1 $. Extend the function $f(x)$ (1) As an odd periodic function with period $2$ (2) As an even periodic function with period $2$ (3) ...
0
votes
1answer
27 views

Stability of a Specific Hill's Equation

Consider the Hill's Equation $u''+a(t)u=0$ where $a(t)=a(t+T)$ for all $t$. Show that if $a(t)<0$ for all $t$, then the solution satisfying the inital condition $u(0)=u'(0)=1$ is unbounded as $t ...
0
votes
3answers
92 views

Why is a sequence $(x_1, x_2, x_3,…)$ eventually periodic if the set $\{x_1, x_2, x_3,…\}$ is finite?

"A sequence $(x_1, x_2, x_3...)$ is eventually periodic if the set ${x_1, x_2, x_3,...}$ is finite. (Think about it.)" (Vivaldi) First of all I am not sure whether this is an if and only if statement ...
0
votes
0answers
33 views

Explicit formula for inclusion/exclusion

I've been searching for a formula for the cardinality of the union of $n$ sets but all the formulas I can find incorporate the symbol (...) and summations that have limits of the form ...
1
vote
2answers
39 views

Turn aperiodic function to periodic

I got a quick question regarding aperiodic functions. Let's suppose I have an aperiodic function $$ f(x) = \left\{ \begin{array}{l l} \exp(-t) & \quad -2\leq t \leq2,\\ 0 & \quad ...
1
vote
4answers
49 views

The period of the function $f(x)=a\cdot \sin(ax)+a$

What is the period of the following function $$f(x)=a\cdot \sin(ax)+a, \mbox{ } x \in \mathbb{R}, a>0.$$ How can I find out? Thanks.
2
votes
2answers
178 views

Period of sum of sinusoids

Say I have a sum of two sinusoids like so: $$ Acos(xt+\phi) + Bcos(yt+\delta) $$ How would I find the period? I know that for just one sinusoid the period would be: $$ Acos(xt+\phi) $$ $$ T = 2\pi/x ...
0
votes
1answer
35 views

$f(x)$ is periodic with period p.

Suppose $f(x)$ is periodic with period p and $g(x)$ is periodic with period q. Let $r$ be the L.C.M. of p and q, if it exists. Then show that: If $f(x)$ and $g(x)$ cannot be interchanged by adding a ...
1
vote
1answer
38 views

$f(x)$ is a periodic function with period $T$.

Prove that if $f(x)$ is a periodic function with period T, then the function $f(ax+b)$, where $a>0$, is periodic with period $\frac{T}{a}$. I started with, $$f[(a(x+T/a)+b]=f[(ax+b)+T]=f(ax+b).$$ ...
0
votes
1answer
50 views

Proving the fundamental period of tangent

I'm very new to math and proofs -- so I apologize if my math skills and vocabulary offends you. I have a question that states: Prove that PI is a fundamental period of the tangent function. I need ...
1
vote
0answers
46 views

Help with the system of 13 equations

Homework from electronics class, but since it looks like a math problem to me i decided to look for help here. I am given this composite periodic signal that has 6 harmonics: ...
0
votes
1answer
29 views

Prove that $\exists x_1,x_2 \in \mathbb{R}$ with $|x_1-x_2|=\frac{T}{2}$ such that $f(x_1)=f(x_2)$

Let $f:\mathbb{R} \to \mathbb{R}$ be continuous periodic function with T>0. Prove that $\exists x_1,x_2 \in \mathbb{R}$ with $|x_1-x_2|=\frac{T}{2}$ such that $f(x_1)=f(x_2)$. I don't even know ...
0
votes
1answer
43 views

what is the period of this sinusoidal function funtion

How can we find the fundamental period of this sinusoidal discrete function $x(n) = 10\cos{(\frac{4n\pi}{31}+\frac{\pi}{5}})$ I tried using the formulae $\frac{2π}{\omega}$ and got the answer 31/2, ...
0
votes
1answer
50 views

Period of recurrence relation mod p

For a recurrence relation like $$f(n)=((k-2)*f(n-1)+(k-1)*f(n-2) )\bmod p$$ with initial conditions .This recurrence holds for n>=4 $$\begin{align}f(1)&=k \bmod p\\ f(2)&=k*(k-1) \bmod ...
3
votes
2answers
63 views

How to determine the period of a binary matrix?

As a part of my research project, I encounter a difficulty to compute the period of a binary cat matrix $C$, which is a square matrix of dimension $n$ with the following properties $|\rm{det}(C)|=1$ ...
2
votes
2answers
99 views

Prove Parseval Identity for $f \in C(\Bbb T) 2\pi$ periodic continuous functions

Question: Prove Parseval Identity for $f \in C(\Bbb T) $ $2\pi$ periodic continuous functions $$ \frac{1}{2 \pi} \int_{-\pi}^\pi |f(x)|^2 dx =\sum_{n=-\infty}^\infty |\hat f(n)|^2 $$ Thoughts: We ...
0
votes
1answer
31 views

Periodic Solution of Ode

Consider the homogeneous equation $ y'+a(x) y=0$, a is with period $\xi$ and continuous on $-\infty<x<\infty$. (a) Show that there exist a constant c such that $y(x+\xi)=cy(x)$ for all x and ...
3
votes
1answer
54 views

Prove f is analytic and periodic

Suppose that there are entire functions $\{f_n\}$ so that for all complex numbers $x+iy$ $$\sum_{n=1}^{\infty} |f_n(x+iy)|^{\frac{1}{n}} \leq e^x$$ Show that $f(z)=\sum_{n=1}^{\infty} f_n(x+iy)$ is ...
6
votes
4answers
275 views

Prove that f(x)=g(x)

Show that if $f,g:\mathbb{R}\to \mathbb{R}$ are continuous and periodic and $\lim_{x\to \infty}[f(x)-g(x)]=0$, then $f=g$
3
votes
1answer
98 views

Floquet Theory - Reducing ODE's to Constant Coefficient ODE's?

Am I right in my reading of the bottom of this page by Arnold, where he apparently says that to any system of first order ode's with periodic coefficients one can find a change of variables reducing ...
1
vote
1answer
65 views

Finding period of a periodic function

I am having some trouble finding the period of this function: $$W(\omega) = \frac{\sin[(2N +1)\omega \Delta t / 2]}{(2N + 1)\sin[\omega \Delta t /2]}$$ Here $N$ is an integer, $\omega$ is angular ...
1
vote
1answer
72 views

The difference between two periodic functions converges to zero, is this two functions identical?

If $f(x)$ and $g(x)$ are two periodic functions, that is, $f(x+T_1)=f(x)$ and $g(x+T_2)=g(x)$ for every $x \in R$. Now that $lim_{x\to\infty}(f(x)-g(x))=0$. Conjecture: $f(x) \equiv g(x)$.
1
vote
3answers
144 views

How to prove periodicity of a trigonometric function

$f(x)= \sin(2x)+3\cos(8x)$ Is the function periodic ? What I did is equalize $f(x)=f(x+T)$ and after noting that $\sin(2x)=\sin(2T)=\sin(8x)=0$ and $\cos(2x)=\cos(2T)=\cos(8x)=1$ we get that ...
2
votes
2answers
25 views

What can be said about a function that is odd (or even) with respect to two distinct points?

This question is a little open-ended, but suppose $f : \mathbb R \to \mathbb R$ is odd with respect to two points; i.e. there exist $x_0$ and $x_1$ (and for simplicity, let's take $x_0 = 0$) such that ...
3
votes
1answer
84 views

Let $f: \mathbb{R} \to \mathbb{R}$ be continuous periodic function with period $T>0$

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous periodic function with period $T>0$. Show that there exists an element $x \in \mathbb{R}$ for which $f(x)=f(x+T/2)$. This is what I came up with ...
0
votes
1answer
18 views

What is the period of $f(X) = (A+K\cdot X) \mod N$

I have a function $f$ mapping from integers to integers as follows: $f(X) = (A+K \cdot X) \mod N$ Where $A,K,X,N$ are positive integers. What is the period of the function?
2
votes
2answers
118 views

Fourier Series and periodicity

Let $f$ be a $2 \pi$-periodic piecewise continuous function and let \begin{equation} f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\cos{nx}+b_{n}\sin{nx} \right] \tag{*} \end{equation} ...
1
vote
0answers
24 views

Solutions for ODE with periodic B.C.

I have the following ODE $$-u(x)''=f(x) \qquad u:[0,1)$$ It is smooth and 1-periodic. Assume that I have a solution u(x). How do I prove that: $$u(x)= u(x)+c \quad c \in \mathbb{R} $$ is also a ...
1
vote
0answers
26 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
0answers
48 views

Triangulation in periodically repeated 3D box

I have a set of points in a 3D box (rectangular parallelepiped), which is periodically repeated out of two its opposite sides. How do I find the Delaunay triangulation for this set of points? Here is ...
1
vote
1answer
57 views

How do I prove that $\cos(\frac{1}{2}x)$ is a periodic function?

Given: $f(x)=\cos(\frac{1}{2}x)$. Prove: f is a periodic function with period 4π My math teacher never went over this so I don't know where to start or what to do :/