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

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What is the period of these functions?

I have two functions as follows: $x = (a-b) \cdot \cos(t) + b \cdot \cos(t\cdot(k-1))$ $y = (a-b) \cdot \sin(t) - b \cdot \sin(t\cdot(k-1))$ What are the periods of functions $x$ and $y$? I found ...
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8 views

Square wave in the limit of infinite frequency

What is the new function obtained when one takes the limit of the square wave function when the frequency is taken to infinity? Does it depend on how the function is written down (e.g. defined as ...
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0answers
8 views

Why is there periodicity in the output of Richard Voss' fractional Brownian motion?

I am trying to figure out why the output of fractional Brownian motion (fBm) as described by Richard Voss (Random fractal forgeries. In: Fundamental Algorithms for Computer Graphics, R. A. Earnshaw ...
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34 views

how to get the solution?

I take the next exercise from Apostol's book. I thought for a while , but nothing. Then I read the solution. I try to think: how did he get it , but nothing. I want to know how to get to the solution. ...
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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 ...
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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? ...
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5answers
59 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
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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
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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.
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1answer
40 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 ...
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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 ...
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1answer
26 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 ...
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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
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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} ...
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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 ...
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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 ...
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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
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1answer
47 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 ...
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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 < ...
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1answer
86 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 ...
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0answers
18 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 ...
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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 ...
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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 $p/k$.
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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 ...
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1answer
26 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?
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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. ...
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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 ...
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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 $x \in \mathbb{R}$. ...
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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
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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
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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 ...
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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 \ ...
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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 ...
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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 ...
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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 ...
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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: ...
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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). ...
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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 $s,t$ $$ ...
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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 ...
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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?
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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 ?
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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 $y=\sin2x$ $\sin2x= \sin2(x+T)$ ...
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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 ...
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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.
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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
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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, ...
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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 ...
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0answers
38 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
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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 ...