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

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1answer
47 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
52 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
23 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 $a(t)$ ...
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0answers
20 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
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1answer
54 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
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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
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2answers
83 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 ...
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0answers
23 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 ...
6
votes
5answers
268 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 ...
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0answers
46 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 ...
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1answer
31 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 ...
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votes
3answers
100 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
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0answers
44 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
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2answers
61 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
59 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
464 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
52 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
69 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
76 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 ...
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0answers
48 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
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1answer
31 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
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1answer
49 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
61 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 ...
0
votes
1answer
40 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 ...
2
votes
2answers
122 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 ...
3
votes
1answer
59 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 ...
3
votes
2answers
93 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 $|\rm{det}(C)|=1$ ...
4
votes
1answer
144 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
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1answer
108 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 ...
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3answers
359 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
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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
105 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
139 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
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0answers
26 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 ...
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0answers
52 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 ...
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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 ...
2
votes
2answers
480 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) ...
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1answer
60 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 :/
1
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1answer
87 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: $f(x) \equiv g(x)$.
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1answer
50 views

Simple question about complex $e^{i}$ and angles

I'm working with angles. I have a hard time figuring something. In electric physics, I have an equation describing an AC voltage function, this way $V_{x} = 0.0469 \cdot e^{-j \cdot 1.083}\cdot ...
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0answers
31 views

problem about Period

we have a, b, m initially where 0 <= a, b < m <= 10^7. And we define a[1] = a, ...
1
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1answer
188 views

fundamental period

$f:\Bbb R\to \Bbb R$ is continous and nonconstant function. Let $p$ be a positive real number such that $f(x+p)=f(x) $ for all $x \in \Bbb R$ . Then there exitst $n\in\Bbb N$ such that ${p \over n ...
1
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1answer
26 views

What functions model this relationship?

I'm currently working a bit on an AI, and in order for it to function, it must be able to quickly predict where a point will be in space, given any distance. The movement of this point may be modeled ...
4
votes
1answer
95 views

$f_1, f_2 : \mathbb{R} \rightarrow \mathbb{R}$ nonconstant, continuous, with period $1, \sqrt{2}$, respectively, then $f_1 + f_2$ is not periodic

I've been working on this problem for several hours, but I keep getting stuck. Suppose $f_1, f_2 : \mathbb{R} \rightarrow \mathbb{R}$ periodic with period $1, \sqrt{2}$, respectively, and that each of ...
1
vote
1answer
163 views

Prove the included formula relating cos(nx) and cos(x)

I'm struggling with the below problem. Can anyone shed some light on it? Show that the below formula is a correct relation between $y = \cos n\theta$ and $x = cos \theta$ for all $n$: $$ x = \frac 12 ...
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1answer
127 views

Is this function periodic? [closed]

Is the following function periodic? $$f(x)=\cos(x)*\cos(x\sqrt5)$$ A function $f$ is said to be periodic with period $P$ ($P$ being a nonzero constant) if we have $$f(x+P) = f(x)$$ for all ...
3
votes
1answer
210 views

Minimum period of function such that $f\left(x+\frac{13}{42}\right)+f(x)=f\left(x+\frac{1}{6}\right)+f\left(x+\frac{1}{7}\right) $

Let $ f$ be a function from the set of real numbers $ \mathbb{R}$ into itself such for all $ x \in \mathbb{R},$ we have $ |f(x)| \leq 1,f(x)\neq constant $ and ...
0
votes
1answer
49 views

Periodic extensions and continuity

Consider the periodic extension $g$ of $f(x)=x^2-1$ where $x \in (0,1)$. Now suppose that $h(x)$ is a continuous function, does $h$ composed with $g$ have to be continuous? I don't think it should, ...
1
vote
1answer
45 views

Periodicity of a function

If f(x) is periodic with period a, would f(tx) be periodic with period a/t? Would f(tx+b) make still have period a/t? Im inclined to think so, because this works for the trig functions, but i'm not ...