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Questions tagged [periodic-functions]

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
89 views

If $f(x+1)=f(x)$ then?

Let $f: \ \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x+1) = f(x)$, $\forall x \in \mathbb{R}$. Then which of the following statement(s) is/are true? $f$ is bounded. $f$ is bounded ...
0
votes
1answer
15 views

What is the period of this signal?

Below is the signal : $y[n] = j ^ n$ Someone told me that the period is 4 ,but he didn't explain me why. Can anyone help me ?
1
vote
1answer
33 views

Integrating a nested periodic function

Suppose $g(x)$ is a differentiable, real-valued, periodic function with period $a$ such that for all $u$, $\int_u^{u+a} g(x)dx=0$. Is it true, then, that $\int_u^{u+a}g(x+g(x))dx=0$ for all $u$?
0
votes
2answers
63 views

Show that $\lim_{n\to\infty} \int_{0}^{1}f(x)g(nx)dx = 0$ $g$ periodic on $\mathbb{R}$

Let $g$ be a continuous periodic function on $\mathbb{R}$ with $g(x + 1) = g(x)$. Assume that $\int_{0}^{1}g(x)dx = 0$. (a) Let $f$ be continuous on all $\mathbb{R}$ Show that $\lim_{n\to\infty} \...
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votes
2answers
62 views

How can I prove that this function is periodic :

I know that a periodic function satisfies $F(x+a)=F(x)$ $y=\left\lfloor\frac{\sin x}x\right\rfloor$ And the period of this fuction is $2\pi$ But putting $f(x+2\pi)$ I can't evaluate denominator ...
0
votes
0answers
18 views

Find the periodicity with the help of Laplace transform

I have a function $$x(t) = \pi\cos(21\omega_0t)+0.1\cos(39\omega_0t)$$ that I want to solve T from the periodicity identity, $x(t)=x(t+T)$. What I have tried now is basically just solving $x(t)=x(...
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0answers
14 views

trapezoid approximation of sinus function

I want to find a reasonable good periodic trapezoid function approximation for the sinus (or cosinus) function. My use case is it to divide the year in 4 epochs: visible light length stays roughly ...
0
votes
0answers
24 views

Given a function which is both periodic and continuous, show there exists the smallest period. [duplicate]

We are given a function $f$ which is periodic and continuous. We also know that $f$ is not a constant function. We need to show that there exists a period that is the smallest period. I tried this ...
1
vote
1answer
75 views

Find all functions satisfying $f(x+1)=\frac{f(x)-5}{f(x)-3}$

Find all functions satisfying $$f(x+1)=\frac{f(x)-5}{f(x)-3}$$ My try: We have $$f(x+1)=1-\frac{2}{f(x)-3}$$ Letting $g(x) =f(x+1)-3$ We get $$g(x+1)=-2-\frac{2}{g(x)}$$ Any clue here?
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0answers
43 views

How To solve this type of inequation

Let $f$ be a regular $2\pi-$periodic function . How to find the functions that solve this inequality $a \le f+f^{'}+f^{"} \le b$ where $a,b \in \mathbb{R}_{+}$ . Could you give me some books to ...
0
votes
1answer
42 views

Calculating a periodic signal (way of solving this)?

I created my own examples so i can have the gist of how to solve the real ones that my homework needs so here we go: $$x(t)=\sum_{n=-\infty}^\infty \Pi\left({t-4n\over2}\right) + \sum_{n=-\infty}^\...
0
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2answers
29 views

A question based on finding the periodicity of function. [closed]

Find the periodicity of the function $f(x+1)+f(x-1)=\sqrt2 f(x)$. I tried to solve the problem by replacing $x$ with $(x+1)$ then $(x+2)$ and $(x+3)$. But could not get any answer please help me out ...
1
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0answers
44 views

Showing $\sin(z)$ is elliptic via conformal maps

One could use Euler's identity to show that $\sin(z)$ is an elliptic function, however, if we define the $\arcsin(z)$ as $$\arcsin(z) = \int_0^z \frac{1}{\sqrt{1-w^2}}dw$$ we would like to show ...
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votes
0answers
11 views

Intersection of periodic unions of intervals

Let $S_1,\dots, S_n$ be unions of intervals with period $w_i$: $S_i = \bigcup (a_i+w_ik; b_i+w_ik), 0<b_i - a_i <1, k \in Z$ Under what conditions can I say that these $n$ sets have non-empty ...
0
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0answers
7 views

Give the fundamental period of the resulting extensions and evaluate each at the given point .

Graph the even and the odd periodic extensions of the given function .Give the fundamental period of the resulting extensions and evaluate each at the given point . I graphed the extension even and ...
0
votes
1answer
9 views

Find the “region of interest” of an unknown function

Given an unknown function $f:\mathbb{R} \rightarrow \mathbb{R}$, is it possible to find it's region of interest? By that I mean either the range in which $f$ does not converge or diverge, e.g. $f(x)=(...
0
votes
1answer
32 views

How to show that a function is periodic?

I have this function $y(t)=6\sin{t}-6+\frac{a}{e^{\sin(t)}}+\frac{6}{e^{\sin(t)}} , a\in{\mathbb{R}}$ where i'm trying to show that its periodic. I have attempted to show that $y(t)=y(t+P)$ , here i ...
1
vote
0answers
38 views

Sharkovsky's Theorem and Triangular Functions

I'm trying to prove that Sharkovsky's Theorem Let $\vartriangleleft$ denote the Sharkovsky ordering given (informally) by $\underbrace{1\vartriangleleft 2 \vartriangleleft 4\vartriangleleft 8\...
2
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0answers
61 views

Homeomorphism in compact two dimensional manifold, periodic points, and Euler Characteristic.

I want to prove that if a homeomorphism (a continuous bijection with continuous inverse) in a two dimensional manifold doesn't have a periodic point, then the Euler Characteristc of the manifold is ...
3
votes
1answer
96 views

Stability of Mathieu equation: $x''(t)+\cos t \,x(t)=0$

The equation $$ x''(t)+\cos t \,x(t)=0 \quad (1) $$ can be transformed to the system: $$\vec{x}'= \begin{pmatrix} 0 & 1\\ -\cos t & 0 \end{pmatrix} \vec{x}=A(t) \cdot x(t) $$ with minimum ...
3
votes
7answers
96 views

$f$ is T periodic and $f(x) + f'(x) \ge 0 \Rightarrow f(x) \ge 0$

Let $f: \Bbb R \to \Bbb R$ be a function such that $f'(x)$ exists and is continuous over $\Bbb R$. Moreover, let there be a $T > 0$ such that $f(x + T) = f(x)$ for all $x \in \Bbb R$ and let $f(x) +...
2
votes
0answers
48 views

Show that this system has at least one unbounded solution as $t \to \infty$

Assume the system $$x'(t)=\begin{pmatrix} \frac12-\cos t & 2 \\ 1 & \frac32+\sin t \end{pmatrix}x(t)=A(t)\cdot x(t)$$ with minimum period: $T=2\pi$. Let $\mu_1,\mu_2$ be its characteristic ...
0
votes
1answer
35 views

Why the periodicity of solution for $\theta''+\gamma\theta = 0$ implies $\sqrt{\gamma} = n\in\mathbb{N}$?

One solution for $$\theta''+\gamma\theta = 0$$ is, for $\gamma>0$, $$\Theta(\theta) = A\cos \sqrt{\gamma}\theta + B\sin \sqrt{\gamma}\theta$$ My book says that because of the $2\pi$ periodicity ...
3
votes
0answers
47 views

Is the function $f(x)=\frac{\sin(\sin{(x)}+x)}{2+\cos{(\lvert x\rvert}+\cos{(x)})}$ monotonic and/or periodic?

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ $$f(x)=\frac{\sin(\sin{(x)}+x)}{2+\cos{(\lvert x\rvert}+\cos{(x)})}$$ I am having trouble showing whether or not this function is monotonic and/or periodic. ...
1
vote
2answers
43 views

How to initialize this process to make it produce a sequence with specified period?

Let $S=(1,1,a)$. Then construct a new, infinite sequence $X$ using the following process: First, $X_0=S_0=1$ Then add $S_0$ to each element of $S$, with letters being converted to their position in ...
0
votes
1answer
32 views

integral f(-x) from -a to a equal to integral f(x) from -a to a

Can someone help with a short algebraic proof that $\int_{-a}^ag(x)=\int_{-a}^ag(-x)$ From making a sketch this seems to be correct and you could argue from the graph that it would be correct as the ...
1
vote
1answer
27 views

Period of $|\sin(\pi t)|$ - rectified wave, Fourier Series

I have a problem where I have to find the fundamental period and frequency of the following rectified periodic function $$ x(t) = |\sin(\pi t)| $$ From my understanding I know that the fundamental ...
3
votes
2answers
76 views

Is $\cos(\frac x6) \cdot \cos( \frac {x \cdot \pi}{6})$ periodic?

Aim : To obtain the period of $\cos(x/6)\cos(x\pi/6)$, if it exists What I've done as of now : $$\cos\left(\frac x6\right)\cos\left(\frac {x\pi}{6}\right) = \frac{1}{2} \cdot \left[\cos\left(\frac {...
2
votes
0answers
45 views

Can the solution to this Laplace problem be analytically continued outside of the domain?

Consider the subset $\Omega$ of $\mathbb{R}^2$ which is bounded by the vertical lines $x = 0$ and $x = L$, the horizontal line $z = -h$ and the curve $(x, \eta(x))$. In the figure below, $\Omega$ is ...
1
vote
1answer
55 views

Take $f$ and entire function such that $f(z) = f(z+1)=f(z+\sqrt{2})$, then it is constant [duplicate]

I am working through some practice problems and I have this one which is stumping me: Take $f$ and entire function such that $f(z) = f(z+1)=f(z+\sqrt{2})$, then it is constant. I was thinking, given ...
0
votes
1answer
23 views

Periodic boundary condition confusion

If we have a differential equation on [a, b], I know that a periodic boundary condition is written like $$f(a)=f(b)$$ in my book. I am confused why I am being told in classes that $$f(a)=f(b)=0$$ is ...
2
votes
2answers
38 views

Period of product of two complex exponential functions

I have read that that period of sum/product of two periodic functions is the least common multiple of there individual time periods if the time periods are rationally related but I am not able to ...
0
votes
0answers
12 views

If f is integrable on an interval [a,a+T] of length T,then prove that f is also integrable on any other interval [b,b+T] of length T.

If I can show that the function is integrable on any other interval of length T,I can certainly show that the integration values of that function over these two different intervals of same length are ...
4
votes
1answer
40 views

Periodic solutions of the double pendulum

I'm stuck: Are there periodic solutions of the double pendulum, or not? The question is four-fold: Does every double pendulum - defined by two masses $m_1$, $m_2$ and lengths $l_1$, $l_2$ - have ...
2
votes
1answer
32 views

Floquet theorem in mathematics and physics

I am using the notation from Proofwiki. The Floquet theorem states that for a continuous matrix function $A(t)$ with period $T$ and a fundamental matrix $\Phi(t)$ of the system $x'(t)=A(t)x(t)$, it is ...
4
votes
0answers
173 views

Periodicity criteria for $f'(x)=P^{m}(f(x))$

$$f'(x)=P^{m}(f(x))$$ ,where $P(x)$ is a polynomial, m is a real number. $x\in\mathbb{C}$ and $f:\mathbb{C}\to\mathbb{C}$ I would like to find a criteria to define if $f(x)$ is a periodic function ...
1
vote
0answers
44 views

The functional equation $f(-x+b)=f(x)$

I can solve the (periodic) functional equation $f(x+b)=f(x)$ completely ($x\in \mathbb{R}$ and $b\neq 0$). Indeed, its general solution is $f=\phi o (\; )_b$, where $(\; )_b$ is the $b$-decimal (...
0
votes
1answer
25 views

If the limit of the subtraction of two periodic function is ZERO when x goes to INFINITY, is these two functions equals to each other? [closed]

If $f:\Bbb R\to \Bbb R$ and $g:\Bbb R\to \Bbb R$ are periodic functions such that $$\lim_{x \to + \infty }(f-g)(x)=0$$ Prove $f=g$, or give a counter example. By the way, please notice that the ...
5
votes
1answer
122 views

Must a continuous and periodic functions have a smallest period?

Let $D\subset\mathbb R$ and let $T\in(0,+\infty)$. A function $f\colon D\longrightarrow\mathbb R$ is called a periodic function with period $T$ if, for each $x\in D$, $x+T\in D$ and $f(x+T)=f(x)$. ...
3
votes
1answer
65 views

Sharkovsky's theorem, period 4 implies period 2

I need to prove that using the ordering of the Sharkocsky's, that period 4 implies period 2. Thus for a continuous function f from the unit interval to the unit interval itself, I need to prove that $...
2
votes
4answers
86 views

can the sine of a polynomial be periodic?

For which polynomials $p(x)$ is $\sin p(x)$ periodic? I started by observing that if $f(x)$ is periodic than also $f'(x)$ is periodic. Than $p\,'(x) \cos p(x)$ is periodic. Of course if $\deg p\,'(x) ...
0
votes
1answer
36 views

A product of two functions is periodic; are the functions individually periodic?

I'm interested in the converse of the question here: Period of the sum/product of two functions. Instead of "given two periodic functions, $f(x)$, $g(x)$, what is the period of a sum $f(x)+g(x)$ or ...
9
votes
5answers
1k views

Function with arbitrary small period

Is there a function f: $\mathbb{R} \to \mathbb{R}$ with arbitrary small period different from $f(x) = k$? ($\forall \epsilon >0 \exists a < \epsilon $ such that f(x) has a periodicity $a$) I ...
0
votes
2answers
23 views

Calculating periodicity of general function

I have a function which is written as $$ \left\|r_1+\frac{(1-r_1^2)r_2e^{-i\delta}}{1-r_2^2e^{-i\delta}}\right\|^2 $$ where $r_1, r_2$ are constants, and $\delta$ is variable. This is originally from ...
0
votes
1answer
78 views

If $f$ is $2\pi$-periodic and $C^{k}$, $\hat{f}(n)=O\left(\frac{1}{|n|^{k}}\right)$ as $n\rightarrow\infty$

My question is: Suppose that $f$ is $2\pi$-periodic, and $C^{k}$. Show that $\hat{f}(n)=O\left(\frac{1}{|n|^{k}}\right)$ as $n\rightarrow\infty$. This notation means that there is a constant $C$ such ...
1
vote
2answers
55 views

The Existence of a Periodic Solution in a Non-Homogeneous System of ODE's

Variation of parameters tells us that the solution to the equation $$\dot{X}(t) = AX(t) + G(t)$$ with initial condition $X(0) = X_0$ is $$X(t) = \exp(tA)(X_0 + \int_0^t \exp(-sA)G(s)ds)$$ where $A$ ...
0
votes
1answer
111 views

Is Heaviside step function or unit step function periodic?

I have a unit(or Heaviside) step function in discrete form: $$\text u[n]=\begin{cases} 0, & n < 0, \\1, & n \ge 0, \end{cases}$$ and in continuous form: $$\text u(t)=\begin{cases} 0, &...
2
votes
2answers
63 views

Main period of $f(x)=\cos 5x+\cos 10x$

Here is what I have done so far: \begin{align*} f(x)&=\cos5x+\cos10x\\ f(x)&=\cos5x+2\cos^2(5x)-1\\ f(x)&=2\cos^25x+\cos5x-1\\ \end{align*} I have tried to further simplify the function to ...
0
votes
1answer
63 views

Is the set of all periodic functions with similar period, a vector space?

Let $S$ be the set of all the periodic function with period $T$, is it a vector space? I know a vector space is a set that is closed under finite vector addition and scalar multiplication, I want to ...
0
votes
1answer
50 views

Period of periodic functions

If I have a function $f(x) = \cos(x) + \sin(x)$ from graphing software, I know the period is $2 \pi$, but can that be shown algebraically? I understand why $\cos(x) = \cos(x + 2 \pi) $ and why $\sin(x)...