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

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0
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2answers
39 views

Find periodicity of a function $f(x)=\frac{\sin x}{2+\cos x}$

$$\exists p\neq 0 : x\in D (f(x+p)=f(x))$$ where $D$ is the domain of $f(x)$. $$f(x+p)=\frac{\sin (x+p)}{2+\cos (x+p)}=\frac{\sin p\cos x+\sin x\cos p}{2+\cos p\cos x-\sin x\sin p}$$ $$\frac{\sin ...
2
votes
1answer
24 views

Is there a family of functions that includes triangle, sin, and square waves?

Is there a family of functions that includes triangle, sin, and square waves? ]2 If so, is there a way to parametrise them such that a single parameter sweeps from triangle through sin to square? ...
6
votes
5answers
58 views

How do we conclude that $f(x)=0, \forall x\in \mathbb{R}$ ?

Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ is a periodic function such that $\displaystyle{\lim_{x\rightarrow +\infty}f(x)=0}$. I want to show that $f(x)=0$ for all $x\in \mathbb{R}$. $$$$ ...
2
votes
1answer
40 views

Find all Periodic function

Can you help me with one question? Find all twice continuously differentiable $2\pi$ periodic functions which: $e^{ix} f''(x)+5f'(x)+f(x)=0$ probably has something to do with Fourier series Any ...
2
votes
0answers
25 views

Limit of sum of periodic function

Let $f_1,f_2,...,f_n$ are periodic functions,if $\lim\limits_{x\rightarrow\infty}\sum_{i=1}^n f_i(x)$ is existent and bounded. How to show $\sum_{i=1}^n f_i(x)\equiv C$ ? $C$ is a constant.
0
votes
1answer
39 views

Proving ODE is periodic if and only if p(t) is periodic

Consider the first-order nonautonomous equation $x′ = p(t)x$ where $p(t)$ is differentiable and periodic with period $T$. Prove that all solutions of this equation are periodic with period $T$ if and ...
0
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0answers
25 views

Is it true that a cycle with a period of 29 hours over 24 hours leads to a non-recurring pattern and how to prove it?

The default 'reset time' for Internet Information Services is 29 hours. The reason for this is that 'Wade [person on the team who developed the setting] suggested 29 hours for the simple reason ...
0
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1answer
25 views

Bound for function with constant /periodic second derivative

Consider a function $f : \mathbb{R}\to\mathbb{R}$ with $f''$ continuous and $f''(x)=f''(x+1)$ for all real numbers x. I need to show that there exists a real positive number $c$ such that $f(x)\leq ...
0
votes
1answer
22 views

Why are the Fourier's coefficient on $0,2\pi$ and $-\pi,\pi$ the same?

I was given the Following Fourier's coefficient and I was happy with it: $$\left\{ \begin{array}{ll} a_n(f)=\frac{1}{\pi} \int_{0}^{2\pi} f(x) \cos\left(nx\right)\,\mathrm{d}x\\ b_n(f) = ...
5
votes
1answer
63 views

Periodic orbits on centre manifold

I am interested in periodic orbits of mechanical systems of second-order dynamics with no damping, i.e. governed by an equation of the type \begin{equation}(1)\quad \ddot x + f(x)=0 \end{equation} ...
1
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0answers
21 views

How to find out the Power of $x(t)$?

I am studying signals and system. I learned that \begin{align} P&=\lim_{L\to\infty} \frac 1{2L} \int_{-L}^{L} |x(t)|^2 dt\\ P&=\frac 1{T} \int_{<T>} |x(t)|^2 dt ~~~\mbox{, P could be ...
0
votes
0answers
25 views

Strong period of self-convolution of a strongly periodic cyclic function

Let $f : \mathbb{Z}_N \to F$ be a function with a $F$ a field. We say that $f$ has maximum period if the smallest positive integer $r$ (with $r \mid N$) such that $f(j) = f(j + r)$ for all $j \in ...
1
vote
2answers
46 views

How to check the following system on existence of periodic solutions?

Let's have following system of DE: $$ \begin{cases} \dot{x} = y(1+x - y^{2}) \\ \dot{y} = x(1+y-x^{2})\end{cases}, \quad x, y \geqslant 0 $$ How to check whether this system contains periodic ...
2
votes
1answer
46 views

Ways to generate triangle wave function.

I recently when searching for parameters on a unit cube in $\mathbb{R}^9$ (we all have our more or less peculiar hobbies, don't we?) found a practical reason to implement a triangle wave function ...
-1
votes
1answer
30 views

Periodic product of sinusoids

(This is problem P-3.7 from the book 'Signal processing first') Let $x(t) = 2\cos(\omega_1t)\cos(\omega_2t) = \cos([\omega_1 + \omega2]t)+\cos([\omega_2 - \omega_1]t)$ where $0 < \omega_1 < ...
1
vote
1answer
85 views

Prove that 012345678910111213 etc is not a periodic sequence.

Prove that the sequence $012345678910111213...$ (all non-negative integers written one by one in natural order) is not periodic. I want to know the shortest and most elegant way to prove it. Can you ...
1
vote
0answers
17 views

Relationship between the integral of a periodic function on the unit circle and an infinite sum.

I am studying for my final and am stumped on this problem. Can someone give me hints or post a detailed solution? Suppose that $f$ is a continuous function on $\mathbb{R}$, with period $1$. Prove ...
1
vote
1answer
14 views

How to find period of a sum of periodic functions

I got this function: $$ x[n]=\sin(2*\pi*4/3*n) + \cos(2*\pi*5/2*n) $$ It is easy to see that period of the sin is 3/4 and the ...
-1
votes
1answer
47 views

Prove that the Dirichlet function is a periodic function with no definite period

Prove that the Dirichlet function $f$ below is a periodic function with no definite period. $$f(x)=\begin{cases}1 ,& x \,\ \text{is rational} \\ 0 ,& x \,\ \text{is ...
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1answer
21 views

Change of scale of periodic function

Could someone show how to prove the following: If $f(x)$ has a period of $p$ show that $f(kx)$ has period of $p/k$.
2
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0answers
169 views

Is $a_n = f(a + b c^n)$ when $a_0 = \ln(2)$ and $a_{n+1} = (e+1) a_n (a_n - 1)$?

Definition Let $a_n$ be a real sequence. Assume there exists a continuous real-periodic function $f(x)$ such that $f(n) = a_n$ And $f(x)$ has the period $t$ , where $t$ is An irrational real ...
0
votes
1answer
25 views

Hot to show that system of nonlinear differential equations doesn't have periodic solutions?

Suppose we have nonlinear system of differential equations $$ \frac{d\mathbf x}{dt} = \hat{A}(\mathbf x, \mu)\mathbf x $$ How to show that it has periodic solutions?
2
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0answers
16 views

Non linear regression calculator

Are there any really good non linear regression calculators around the web? Or is something like matlab the best solution? I tried using excel and its solver tool, but it's complete garbage lol. ...
0
votes
1answer
14 views

Periodic modular piecewise function

For every positive integer $n$, let $\text{mod}_5 (n)$ be the remainder obtained when $n$ is divided by 5. Define a function $f: \{0,1,2,3,\dots\} \times \{0,1,2,3,4\} \to \{0,1,2,3,4\}$ recursively ...
1
vote
0answers
31 views

Proof for a periodic function

I have to solve the following exercise: The function $f : \mathbb{R} \rightarrow \mathbb{R}$ is a periodic function with $P = 2\pi$ so that $f(x) = f(x + 2\pi)$ is true for all $x \in \mathbb{R}$. ...
0
votes
0answers
20 views

Getting sum of $2 \pi$ periodic function

I have a $2\pi$ periodic function which in the interval $[0,\pi]$ is $f(t) = \sin{\frac{t}{2}}$. I have to find the sum for $t \in \mathbb{R}$. But do I know anything about $f(t)$ outside of $t \in ...
1
vote
1answer
57 views

maximum Difference between two zeros

What is the maximum difference between the two consecutive zeros of the solutions of $y''+(1+x)y=0$ on $0\leq x<+\infty$? I have applied the Strum's comparison theorem (by comparison with ...
2
votes
1answer
21 views

Periodic Poincaré Inequality?

The classical periodic Poincaré inequality states that if $u\in H^1(\mathbb T^n)$ is such that $\displaystyle\int_{\mathbb T^n} u(x)\ dx=0$ then $$\|u\|_{L^2(\mathbb T^n)}^2\leq C_d \|\nabla ...
0
votes
0answers
14 views

Solve for a Fourier series problem

I am learning Fourier series and I have a problem which has me confused and would like to here others take on it. $$F(t)=\begin{cases}v_0 & \ \ 0 \le t\le T\\ 0 & \ \ T\le t \le \ ...
1
vote
1answer
26 views

Let $f(x)=\sin^{23}x-\cos^{22}x$ and $g(x)=1+\frac{1}{2}\arctan|x|$,then the number of values of $x$ in the interval $[-10\pi,20\pi]$

Let $f(x)=\sin^{23}x-\cos^{22}x$ and $g(x)=1+\frac{1}{2}\arctan|x|$,then the number of values of $x$ in the interval $[-10\pi,20\pi]$ satisfying the equation $f(x)=\text{sgn}(g(x)),$ is ...
0
votes
0answers
59 views

Examples of real orbits with irrational period

Im looking for examples of periodic orbits on the real line given by piecewise analytic functions $g$. More specific at most 2 pieces. Im talking about integer iterations starting at $f(0)=0$ and with ...
1
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0answers
21 views

Floquet Theory : zero equilibirum stability

Wondering if someone could explain exactly what it means when someone says "zero equilibirium is unstable because the floquet multiplier is greater than 1". I understand the FM part but I don't know ...
0
votes
2answers
18 views

On Laplace transform of periodic functions

I recently bumped into this theorem regarding the Laplace transform of periodic function: Given a periodic function $f(t)=f(t+p)$, where $p$ is its period, then its Laplace transform is given by: ...
1
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0answers
31 views

Suppose that a sequence {$a_n$} is eventually periodic. Prove that {$b_n$} is eventually periodic, given that they are in some relation.

To clarify, a sequence {$x_n$} is called eventually periodic if there exist positive integer $r$ and $p$ for which $x_{n+p}=x_n$ for all $n\geq r$. The relation they have is that, (i). ...
3
votes
2answers
53 views

Property of function $\varphi(x)=|x|$ on $\mathbb{R}$

Define $\varphi(x)=|x|$ on $[-1,1]$ and extend the definition of $\varphi(x)$ to all real $x$ by requiring that $\varphi(x+2)=\varphi(x).$ How do you prove that for any $s,t$ $$ ...
0
votes
0answers
30 views

Does linear combination of periodic functions is periodic?

Does linear combination of periodic functions is periodic? I know that if $f$ is periodic and $g$ is periodic then $f+g$ is periodic. But what about the linear combination of two periodic ...
0
votes
1answer
94 views

If $g(x):=f(x^2)$ is uniformly continuous, is $f$ a constant (assuming it's periodic)?

I know that $f:\mathbb{R}\to\mathbb{R}$ is continuous and periodic. How can I prove that $f$ is a constant whenever $g(x):=f(x^2)$ is uniformly continuous?
1
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0answers
27 views

Are there entire functions that are unexpectedly periodic? [closed]

I wonder if there have ever been entire functions discovered/defined that turned out to be periodic and the periodicity was unexpected and surprising to mathematicians ?
1
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1answer
22 views

How to find the period of this trigonometric function

$y$ = $|\sin x|$ I know the period is π by drawing the graph, but I can't prove it. Please use this method we have learnt for other functions. For example $y=\sin2x$ $\sin2x= \sin2(x+T)$ ...
0
votes
1answer
36 views

Integrating over discontinuities

I have the following integral: \begin{align} \int^T_0e^\tau I(\tau+t^0)d\tau. \end{align} In this integral, $I(t)$ is a function with period $T$. At each time $T, 2T, \ldots$, $I$ is increased by a ...
-1
votes
1answer
38 views

If the function $g: R\rightarrow R$ is periodic, then $a_n=g(n)$ is periodic.

If the function $g: R\rightarrow R$ is periodic, then $a_n=g(n)$ is periodic. Is the converse true? Give a proof or counterexample.
0
votes
0answers
18 views

Understanding the relationship between phase and frequency in a particular equation

Context I have the following equation: \begin{align} a&=1-e^{-\omega t/x}+\frac{m}{\sqrt{1+x^2}}\Bigg[\sin(\alpha+\omega t + \varphi)-\sin(\alpha+\varphi)e^{-\omega t/x}\Bigg]. \end{align} This ...
1
vote
1answer
39 views

Formula for periodic sequence with increasing period

The first values of the sequence are: $\lt1, 0, 1, 1, 0, 0, -1, -1, -1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, ...
5
votes
3answers
123 views

What is the period of $f(x) = \sin^4(x)+ \cos^4(x)$?

This is an elementary problem but I'm just not getting the right answer. My reasoning is as follows: The period of $g(x) = \sin^4(x)$ is $\pi$ and that of $h(x) = \cos^4(x)$ is $π$ as well, so the ...
0
votes
0answers
37 views

Let $f$ be a continuously differentiable function which has period 2$\pi$.

Prove that $\lim_{n\to\infty}$,$$\int_{0}^{2\pi} f(x) \sin(nx) dx = 0$$. First, off I don't know what is $f(x)$ and once I know this I can use integration by parts, bound the integral and apply the ...
2
votes
1answer
71 views

Are all solutions of $ f_n (x)^{2n} + f_n ' (x)^{2n} = 1$ periodic?

Let $n$ be a strict positive integer and $x$ is a complex number. Define $f_n(x)$ as one of the solutions to $$ f_n (x)^{2n} + f_n ' (x)^{2n} = 1$$ Where the derivative is with respect to $x$. Why ...
0
votes
1answer
43 views

Prove that a continuous function on $\mathbb{R}$ which is periodic attains its extreme values.

Prove that a continuous function on $\mathbb{R}$ which is periodic attains its extreme values. I don't know how to start this proof, but I know I have to use the extreme value theorem, continuity ...
0
votes
2answers
24 views

Are periodic functions stationary processes, e.g. y=sin(t)?

According to wikipedia, a stationary process (or strict(ly) stationary process or strong(ly) stationary process) is a stochastic process whose joint probability distribution does not change when ...
1
vote
1answer
20 views

Finding $\sin^3\frac{x}{2}\cdot\cos^7\frac{x}{3}$ periodicity

I need to find the period of $\sin^3\frac{x}{2}\cdot\cos^7\frac{x}{3}$ but couldn't find a nice method how? any ideas?
1
vote
1answer
18 views

Constant function and periodicity

If $f$ is constant function then every real number $p>0$ is its period. I was wondering is the converse true, that is: If every real number $p>0$ is period of a function $f : \mathbb R \to ...