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
52 views

Question about a continuous periodic function [on hold]

Consider the continuous and periodic function $f:\mathbb R \rightarrow \mathbb R$ with period $T > 0$ so that $f(x)=f(x+T)$ for any $x$. Question: Prove that there exists a $c$ such that ...
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0answers
11 views

periodic function sine wave problem relate with sound

SinD +SinC = 2Sin (C+D)/2 Cos(C-D)/2 can anyone help me on how to express this equation as single wave in the case that two sine waves those sum makes a sound? Thank You!
3
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3answers
43 views

Fundamental Period of $\tan x \cot x$

What is the period of $\tan x \cot x?$ I was given this question today. What I did was simplify the expression , and it reduced to a constant function. So it had no fundamental period. But my teacher ...
0
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1answer
6 views

How is Dulac's Multiplier selected?

I'm aware that Dulac's (Negative) Criterion states that a system of differential equations of the form $x' = f(x,y), \; y' = g(x,y)$ has no periodic orbits in the plane if we can find some function ...
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0answers
20 views

How to determine the smallest period of a parametric curve?

Consider the polar function $r(\theta) = \sin(3\theta)$, and the parameterization of its graph given by $x = \sin(3\theta)\cos(\theta), \;y=\sin(3\theta)\sin(\theta)$. Upon inspection, one can observe ...
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0answers
11 views

How to determine if a product of periodic functions is periodic? [duplicate]

Consider two periodic functions $f(t)$ and $g(t)$, and suppose that $f(t)$ has a period $T_f$ and $g(t)$ has a period $T_g$. Is $f(t) \cdot g(t)$ necessarily periodic? If so, what is the period? If ...
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0answers
19 views

How to find the fundamental period of the product of two periodic functions? [duplicate]

Say you wanted to find the period of $x(t)=\sin(at)\sin(bt)$ where $a$ and $b$ are real constants. How could you go about this?
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2answers
18 views

Fourier series: can a function be odd and have a dc component?

Long story short: fourier series is taken in two subjects (for now). One doc says that the dc component is 0 if the function is odd. The other says that odd and even has no effect on the dc ...
2
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0answers
85 views

Periodic solution to DDE: $\frac{d}{dt}x(t)+\left(\frac{\pi}{2}+\epsilon\right)x(t-1)[1+x(t)]=0$

Consider differential equation with delay: $$\frac{d}{dt}x(t)+\left(\frac{\pi}{2}+\epsilon\right)x(t-1)[1+x(t)]=0.$$ Let's use $t=(1+c)\tau$ substitution to normalize time t: ...
3
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3answers
85 views

How to find a formula for a periodic sequence?

I would like to find the formula for a periodic sequence such as 4, 1, 1/4, 1/4, 1, 4... with a period of 6
2
votes
1answer
57 views

Integration of periodic functions

I have a question at hand (which may be easy to some) ,but unfortunately I don't know how to even begin with. Could someone help me? If $f$ and $g$ are continuous, $2\pi$ periodic functions then ...
1
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0answers
31 views

Find the number of points of distance n away from origin as function of n

I came across a seemingly simple problem the other day and I thought I'd share it with anyone interested. Say you have a point in 3 dimensions. The number of points that are of distance $0$ away is ...
0
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1answer
13 views

Construct a Sinusoidal Equation for an Irregular Period

I would like to be able to construct a sinusoidal function of limited domain given a set of real roots, assuming that the function is graphically centered on $y=0$. I expect that this would ...
1
vote
1answer
49 views

What is the period of a fuction which satisfies the condition $f(a-x)=f(a+x)$?

What is the period of a function which satisfies the condition $f(a-x)=f(a+x)$ where a is any positive integer? I tried substituting $x$ with $x-a$ but that does not seems to help me a lot. I ended ...
0
votes
1answer
29 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
votes
1answer
23 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) ...
1
vote
1answer
27 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
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0answers
13 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
vote
1answer
19 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
313 views

How to prove that a function is positive

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 ...
2
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1answer
36 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
vote
1answer
27 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
51 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 ...
4
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1answer
104 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
26 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} &= ...
1
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0answers
23 views

Finding the period from a graph to create a function

I'm trying to convert a graph from this website into a function where y = d + A sin (k (x - delta) ) I have already gotten the vertical shift, amplitude and the ...
2
votes
3answers
44 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
24 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
40 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
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2answers
42 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) = ...
6
votes
1answer
61 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 ...
1
vote
1answer
82 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
vote
1answer
34 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
65 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
votes
1answer
40 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
38 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
votes
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
30 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 ...
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0answers
43 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 ...
-1
votes
2answers
62 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? $w_1=(5/2)=2.5 \rightarrow T_1 = 2\pi/w_1 = 2\pi/2.5 =2.513$ $w_2=(1/4)=0.25 \rightarrow T_2 = 2\pi/w_2 = 2\pi/0.25=25.13$ ...
9
votes
1answer
131 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
vote
3answers
39 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
votes
0answers
13 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
36 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
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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 ...
2
votes
1answer
36 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
62 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
42 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
95 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.