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
32 views

My brain doesn't work right now: What's the formula for the $n$th vertex of a discretized sine wave?

So far I have: $$ A \sin(2\pi f ? + \phi) $$ where $f$ is cycles per second, and $\phi$ is in seconds. If I'd like to approximate the sine wave with $N$ points per cycle, and I want to draw $C$ ...
0
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1answer
24 views

Periodic Foricing Terms

The question asks to find the solution for the initial value problem: $ y''+\omega^2y=sin(nt),\quad y(0)=0,\quad y'(0)=0 $ where $n$ is a positive integer when a) $\omega^2\neq n^2$ and b) $\omega^2=...
1
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1answer
29 views

Provinf Orbits are eventually fixed or eventually prime-2-periodic

Please I need help with this question: Let $S = \{a,b,c,d\}$ be a finite set and suppose that $f \colon S \to S$ has the property that: $$f(a) = f(b) = f(c) = d$$ Prove that each of the orbits of ...
0
votes
0answers
22 views

What is the periodicity of an infinite sum of Dirac delta functions?

I have the following function: $$F(\mu)=\sum_{n=-\infty}^\infty [\delta(\mu - n/\Delta T - a) + \delta(\mu - n/\Delta T + a)]$$ Where $\Delta T$ and $a$ are positive real constants. It is a result ...
1
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1answer
21 views

Sum of a periodic sequence of functions

Suppose that $x_j$ is an $n$-periodic sequence. Show that $$\sum_{j=m}^{m+n-1}x_j=\sum_{j=0}^{n-1}x_j.$$ So far I have tried playing around with the indices of the sequence and have \begin{align*} ...
11
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1answer
337 views

Different way to show $\int_{-\infty}^{\infty}e^{-\frac{\cosh^2(u)}{2x}}\,e^{-\frac{u^2}{2 t}}\,\cos\left(\frac{\pi\,u}{2t }\right)\,\cosh(u)\,du > 0$

I have been long time trying to prove that the following integral $$I_t(x)=\int_{-\infty}^{\infty}e^{-\frac{\cosh^2(u)}{2x}}\,e^{-\frac{u^2}{2 t}}\,\cos\left(\frac{\pi\,u}{2t }\right)\,\cosh(u)\,du,\...
2
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1answer
38 views

problem in Functions and Periodicity

What is the period of $f(x+\frac{1}{2}) +f(x-\frac{1}{2})=f(x)$ ? I tried substituting $x=x+\frac{1}{2}$ and $x=x-\frac{1}{2}$ but that didn't get me anywhere. According to the standard procedures , ...
1
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1answer
37 views

show that an ordinary differential equation has T periodic solution

I have $dx/dt=-x^3(t)+h(t)$ where h(t) is a smooth, T-periodic function. Show that $x'(t)$ has a periodic solution. So I tried solving the function as letting $h(t) =sin(t)$ and $cos(t)$ which are $...
2
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1answer
59 views

Layperson's explanation of Euler's formula

A few weeks back I asked a question which lead to Euler's formula being brought up. I don't have the mathematical background to fully appreciate it's purported mathematical beauty. Just yesterday I ...
6
votes
3answers
170 views

Is $(\sin{x})(\sin{\pi x})$ periodic?

I'm wondering, is the function $f=(\sin{x})(\sin{\pi x})$ is periodic? My first inclination would be two assume that if the periods of the individual sine expressions, $p_1 \text{and}\space p_2$ have ...
0
votes
2answers
30 views

Can't get the period of the sum of a sine and a cosine

$$ x(t) = 2\cos(5t+\pi/10) + 3\sin(5\pi t) $$ I'm supposing that the signal is periodic (because sine and cosine are periodic) but then; \begin{align} P_{\sin} &= \tfrac{2}{5} \\ P_{\cos} &= ...
2
votes
3answers
75 views

Constant functions periodic?

I dont understand the meaning of this line in my book - " $\sin^2x + \cos^2x$ is periodic but the fundamental period is not defined. " Why is the period not defined? $F(x)$ is $1$ here so it is a ...
0
votes
1answer
27 views

Period of a trigonometric series

What is the period of the function represented by the series $a_1 cos x+a_2cos2x+a_3cos3x+...$ I guess it is $2\pi$. Am I right?
1
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1answer
45 views

how to find its period?

I have this function $T(x)=x+4B \pmod A$. I want to solve the congruence for the smallest positive $n$, $T^{n} (x)=x \pmod A$. How to solve it and find its period? To solve it, what I did is by ...
1
vote
2answers
71 views

Laplace Transform of Square Wave Function

I am given a problem in my textbook and I am left to determine the Laplace transform of a function given its graph (see the attached photo) - a square wave - using the theorem that $$F(s) = \frac{1}{1-...
6
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1answer
62 views

When does a linear combination of trigonometric functions have an axis of symmetry?

I am trying to find out when a linear combination of $\sin(ax)$ and $\cos(bx)$ has an axis of symmetry. Clearly, $\sin(x)+\cos(x)$ has an axis of symmetry at $\pi/4$. It seems as if $\sin(3 x)+\cos(...
1
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1answer
90 views

How to find periodicity of a nim sequence?

I am trying to solve a problem which is a simple algorithmic game. Link to the problem - https://community.topcoder.com/stat?c=problem_statement&pm=6856 I have basically figured out that for ...
1
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1answer
36 views

period of the function $y = \sin^2\frac{\sqrt2x+3}{6\pi}$

Today I came across a question The fundamental period of the function $y = \sin^2\frac{\sqrt2x+3}{6\pi}$ is $\lambda \pi^2$ then the value of $\frac{\lambda}{\sqrt2}$ is ___ I tried to equate ...
0
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1answer
69 views

How to prove that the ellipse is a periodic orbit knowing that the orbital derivative of a function V is zero on there

The question is as follows: Show that the orbital derivative of the function $V=(1-x^2-2y^2)^2$ is zero on the ellipse $x^2+2y^2=1$, and explain why you can deduce that the ellipse is a periodic ...
2
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1answer
43 views

Proof: A solution to a periodic ODE shifted by a constant time is another a solution to that ODE.

This is probably a trivial question but I don't have a clue where to begin. Suppose $x(t)$ is a T-periodic solution to the differential equation $$\frac{dX}{dt}=F(X)$$ where F(X) is in $C^1$. Show ...
0
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0answers
40 views

True or False: $f(z)=Ln(z)$ is periodic

From Wolfram MathWorld: A function $f(x)$ is said to be periodic with period $p$ if $f(x)=f(x+np)$ for $n=1,2,3...$ The easy bait here is to realize that $Ln(z)$ is asking for the principal ...
2
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1answer
37 views

Are there any theorems linking periodic functions to the number of times they are differentiable?

I was working through some Fourier series questions and I was wondering if the periodicity of a function has anything to do with the number of times it's differentiable. For instance, the elementary ...
0
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1answer
33 views

Compute Limit Involving Integral and Periodic function [closed]

If $f: \bf R \to \bf R$ is continuous and periodic with period $T$, then show that $$ \frac{1}{t}\int_{a}^{a+t}f(s)ds \to \frac {1}{T}\int_{0}^{T}f(s)ds$$ where $a\in \mathbb{R}$ and $ t \to \infty$...
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0answers
44 views

Periodicity of a trigonometric function

How to find the fundamental period of the function $|\sin x - \cos x|$ and $|\sin x - \cos x|$ + $|\sin x + \cos x|$? Please tell the proper method to find the period of the functions like these. I ...
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2answers
78 views

Is $x(t)=\sin(5t/2)+\cos(2t/8)+\sin(3t/6)$ periodic or aperiodic? Find the fundamental period and frequency of the signal.

Is $x(t)=\sin(5t/2)+\cos(2t/8)+\sin(3t/6)$ periodic or aperiodic? $w_1=(5/2)=2.5 \rightarrow T_1 = 2\pi/w_1 = 2\pi/2.5 =2.513$ $w_2=(1/4)=0.25 \rightarrow T_2 = 2\pi/w_2 = 2\pi/0.25=25.13$ $w_3=(1/...
9
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1answer
160 views

Smooth sawtooth wave $y(x)=\cos(x-\cos(x-\cos(x-\dots)))$

Consider an infinite recursive function $$y(x)=\cos(x-\cos(x-\cos(x-\dots)))$$ $$y=\cos(x-y)$$ Plotting the function $y(x)$ implicitly we get a smooth sawtooth-like wave: Was this function ...
1
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3answers
40 views

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 ...
0
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0answers
18 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 $\...
1
vote
0answers
13 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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0answers
37 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. ...
0
votes
2answers
43 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 x}{...
2
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1answer
42 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
63 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
43 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 ...
5
votes
1answer
97 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.
1
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1answer
67 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 ...
2
votes
3answers
78 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
38 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 c(...
0
votes
1answer
27 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) = \frac{2}...
5
votes
1answer
76 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
23 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
37 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
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2answers
79 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
56 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 $t(x)...
-1
votes
1answer
31 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
95 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
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0answers
22 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
27 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
83 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. $$f(x)=\begin{cases}1 ,& x \,\ \text{is rational} \\ 0 ,& x \,\ \text{is irrational}\end{cases}$$ ...
-1
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1answer
43 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 $p/k$.