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
22 views

What function(s) satisifies $f(\theta)=-f(\theta+2 \pi )$?

This may be a trivial question, but perhaps someone can give a detailed answer. I'm looking for a periodic function that satisfies $$f(\theta)=-f(\theta+2\pi)$$ where $\theta$ is an angle in radians. ...
0
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0answers
13 views

Determine Periodicity with imaginary numbers

I have a function $\cos(a\cdot n) + j\sin(a\cdot n)$, for $|n| < 10$ ($a$ is some constant). My textbook says that this function is not periodic; I was wondering why this is? Is it because n is ...
0
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1answer
38 views

Formal solution needed to question that looks too easy to be true about the Gauss map

Using the itineraries of the Gauss map write the continued fraction expansion of the number $0 \leqslant \alpha \leqslant 1$ such that $$\displaystyle \alpha = \dfrac{1}{4+\dfrac{1}{3+\alpha}}$$ I ...
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0answers
22 views

Periodic solutions and critical points

I was going through a lecture, and for an ODE: $x' = x(5-x-2y), y'=y(-6x+x+3y)$ Which has critical points at : $(0,0) (0,2) (3,1) (5,0)$ My professor posed the question as to why the periodic ...
-3
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0answers
17 views

Is these functions periodic ones? [on hold]

Determine whether the function is periodic. If it is periodic, find the smallest (fundamental ) period $f(x)=cos(2x) + 3sin(\pi x)$ $f(x)=sin(2x)-cos(5x)$ thanks in advance
1
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0answers
46 views

Finding the critical values of a response curve

I have the motion of a forced spring: $$x'' + \kappa x' + x - x^3 = \varGamma \cos(\omega t) \ \ \cdots \ \ (1)$$ and I am investigating the stability of its solutions with forcing period $T = ...
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0answers
66 views
+50

Quadratic Irrationality of the Periodic points of the Gauss map

If $G:[0,1] \rightarrow [0,1]$ is the Gauss map which is defined as $$G(x) = \left\{\frac{1}{x}\right\} = \frac{1}{x} - \left\lfloor\frac{1}{x}\right\rfloor,$$ show that if $x$ is periodic of order ...
0
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1answer
13 views

Create a sine function to model the displacement,

When doing another experiment involving a swinging pendulum it was found that the pendulum did 48 complete swings per minute. The distance between the extreme positions of the pendulum was 6.7cm. ...
2
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1answer
46 views

Proving the equality of a sum and integral.

Taken from Rudin's Real and Complex Analysis text: Suppose $f$ is a continuous function on $\mathbb{R}^1$ with period $1$. Prove that $\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N f(n\alpha) ...
0
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0answers
30 views

Minimum number of periods to determine periodicity

Is there a minimum window window length to determine true periodicity with a given estimated period. For example, if the same event is tested for every second and is observed twice at times 1 minute ...
2
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1answer
56 views

If $f$ and $g$ are periodic functions, is $g \circ f$ periodic?

If $f$ and $g$ are periodic functions, is $g \circ f$ periodic? If it is, what is the period? So I know: $f(x) = f(x + T), T \in R$ $g(x) = g(x + P), P \in R$ I have this question for my homework. ...
2
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5answers
37 views

Proving a function $f(x + T)=k\;f(x)$ satisfies $f(x)=a^x g(x)$ for periodical $g$

I need to prove the following: If a function $\,f$ satisfies $$f(x+T)=k\;f(x), \forall x \in \mathbb R$$ for some $k \in \mathbb N$ and $T > 0$, prove that $\,f$ can be written as ...
3
votes
1answer
34 views

Detecting sinus with unknown period

I have some signal source, that can be in one of two states -- it is either emitting constant value 1.0 or oscillating in the way very close to sinus function from ...
1
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1answer
121 views

Is sin(1/x) periodic?with what period time? [closed]

Is sin(1/x) periodic? with what period time?
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2answers
44 views

Did I find the period of $\sqrt{1-\sin^2x}$ correctly?

$f(x)=\sqrt{1-\sin^2x} = \sqrt{\cos^2x} = \vert \cos x\vert$, period is $\pi$. Is this a correct way to find the period of this function? Can I just state that the period of $\vert\cos x\vert$ is ...
1
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3answers
33 views

How to find the period of the function $f(x)= 1/(2+cos(3x))$

I'm supposed to determine whether this function $$f(x)= \frac{1}{2+\cos(3x)}$$ is periodic or not, and find the period if it is periodic. According to my calculator the period is $2π/3$ but I don't ...
6
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2answers
207 views

determine if a function is periodic

Let $f$ be a continuous and integrable function on $[a,b]$ such that $$\int_a^b f(x)\,\mathrm{d}x = 2$$ and for every $t_1,t_2$ such that $\displaystyle t_2 -t_1 = \frac{b-a}2$ ...
1
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2answers
46 views

Periodic Points and One Dimensional Maps Homework Help

Let f be the tripling map $f(x)=3x\mod(1)$. I need to make a table that includes the following for $n\le6$: number of points in Fix($f^n$), number of points in Fix($f^n$) of lower period, number of ...
0
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2answers
69 views

How to determine the orbits of points under the tripling map $f(x)=3x\bmod 1$?

Let $f$ be the tripling map $f(x) = 3x \mod(1)$. Determine the complete orbit of the points $\frac{1}{8}$ and $\frac{1}{72}$. Indicate whether each of these points is periodic, eventually periodic, or ...
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2answers
35 views

Non-trigonometric Continuous Periodic Functions

I've seen lots of examples of periodic functions, but they all have one thing in common: They all involve at least one trigonometric term (e.g. $\sin\theta$, $\cos\theta$, etc.). My question is ...
0
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1answer
15 views

Period of two equal functions

I'm dealing with a problem here. We know that two functions are the same if they have the same domain and codomain. Let's say we have given the functions $f$ ang $g$ where ...
2
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0answers
26 views

Is a horizontal line considered periodic? [duplicate]

Given the following definition of a periodic function: $$\exists P, P > 0, f(x + P) = f(x)$$ It is possible to argue that $f(x)=k$ ($k$ being a constant) is a periodic function, since you can ...
2
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3answers
50 views

In what sense is a function on a circle the same as a $2 \pi$ periodic function on $\mathbb{R}$?

I was reading the appendix of Elias M Stein's Fourier Analysis and before proving the approximation lemma the author mentions the following Recall that a function on a circle is the same as a $2 ...
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0answers
38 views

period of cubic trigonometric functions

Can anybody explain how you would find the period of cubic trigonometric function. so I need to find the period of $f(x)=\sin^2\left(\frac{x}{3}\right)$. So I have began the question by finding the ...
3
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1answer
65 views

The periods of the Weierstrass function $\wp(z)$

Is it true that the periods $\omega_1$, $\omega_2$ of $\wp(z)$ are $\omega_1 = 4K$ and $\omega_2 = 4iK'$, respectively? Here, $K = K(k)$ is the complete elliptic integral of the first kind, and $K' = ...
10
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2answers
191 views

Reversing the process of taking the “sine of an arbitrary shape”

I'm sure we've all seen images such as the following, from wikipedia: link. They give us some nice intuition on what the sine and cosine functions are. Some people may also have seen images such as ...
0
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1answer
39 views

Determining if a sum of trig functions is periodic

Given the discrete-time function $f[n] = 2\cos(\frac{\pi}{4}n) + \sin(\frac{\pi}{8}n) - 2\cos(\frac{\pi}{2}n + \frac{\pi}{6})$ How can I show that the function is periodic? I know that a discrete ...
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2answers
24 views

Finding Principale period of $\cos$ function

Find principle period of $3\cos (2x-3)$. Today I have learned about principle period of various trigonometric function. I know that principle period of cos is $2 \pi$. Please someone can help me ...
3
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1answer
41 views

Find the values of $f(0)$, $f(4)$, $f(6)$ and $f(18)$

A function $f:\mathbb{R} \to \mathbb{R}$ is such that $f(2)=2$ and $$f(x+1)+f(x-1)=\sqrt{3}f(x) \tag{1}.$$ Find the values of $f(0)$, $f(4)$, $f(6)$ and $f(18)$. My approach: replace $x$ ...
1
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4answers
53 views

How to determine the periods of a periodic function?

I am aware of the other similar questions but was not able to figure out what I want to know from those question thus posting it here. Given a periodic function $f(x)=sin(x)$, Why is the period ...
1
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2answers
74 views

Principal period of $\sin\frac{3x}{4}+\cos\frac{2x}{5}$ [duplicate]

Find the principal period of $$\sin\frac{3x}{4}+\cos\frac{2x}{5}$$ It was easy to find principle when single trigonometric function is given, but i don't know how to find principal period of sum of ...
0
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1answer
27 views

Differential Equation and periodicity

Let $f\in C^0(\Bbb{R},\Bbb{R})$. Show that the equation $y''+2y'+2y=f \tag{E}$ has at most one periodic solution. I can solve the equation $y''+2y'+2y=0$ but I am not sure it's useful here. ...
3
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2answers
166 views

Prove that $\sin(\sqrt x)$ not periodic

$\sin\sqrt x$ is not a periodic function. How can one prove this?
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2answers
83 views

Finding the equation for a (inverted) cycloid given two points

If I have two points on a Cartesian plane, and I know that they are connected by a cycloid, then how do I find the equation for that cycloid? For background information, I have been playing around ...
1
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1answer
36 views

Proof of Weierstrass' second theorem using the Fejér operator

Weierstrass' second theorem states the following: Let $f$ be a real continuous $2\pi$-periodic function (write $f\in C_{2\pi}$). Then for all $\epsilon>0$ there exists a trigonometric polynomial ...
3
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1answer
78 views

Making a function periodic

This might not be the best place to ask this question, but here it goes... I'm creating a game and need 3D sea waves. Since it's for mobiles, there's no time to generate entire screen worth of waves ...
4
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2answers
105 views

Why is the period of $f$, $\pi$?

I came across a problem, which asked to compute the period of the function $$f(x)=3^{\sec^2x-\tan^2 x}.$$ The answer provided was $\pi$. I don't get how.
3
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1answer
33 views

Oscillations about equilibrium for coupled differentail equations

I have the following system of equations: $$\begin{align} \frac{dX}{dt} &= 2Y-2\\ \frac{dY}{dt} &= 9X-X^3 \end{align}$$ I would like to study the property of solutions to this function about ...
2
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1answer
95 views

What's the difference between a cyclic and periodic function?

I've seen the words "cyclic" and "periodic" used to describe characteristics of a given function. What do they mean? I can't seem to find a difference. Wikipedia says a periodic function is one that ...
1
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2answers
56 views

Find the amplitude and period of the function. $y = 4 \sin(−6x)$

Do I factor the $-6$ out then divide $2π/-6$ to get the period?
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0answers
19 views
0
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2answers
25 views

determine period of given signal

i would like to compute Fourier coefficients from given signal,and i have following picture i need to know period,just to make sure that i am not making mistake,period should be $\frac {T} {2}$ ...
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0answers
32 views

About the maximal interval of existence

Let $f:\mathbb R\times \mathbb R^n\longrightarrow\mathbb R^n$ be a continuous function such that there exists some $T\in\mathbb R$ with the following property: $$f(T+t,x)= f(t,x)\;\;\forall ...
1
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2answers
68 views

Is signal periodic? What is the period?

Below is the signal : $$ y[n] = \sin\left( \frac{6\pi}{7} n + 1 \right) $$ According to me the Fundamental period is $7/3$ but is the signal periodic? I think it should satisfy this $\sin(6(\pi/7)n ...
2
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0answers
28 views

Entire periodic $f(z)$ with more than 50 % of the derivatives $0$?

Im looking for a real-entire function $f(z)$ such that for any complex $z$ : $1) $$f(z+p) =f(z)$ With $p$ a nonzero real number. $2)$ $f(z)= 0 + a_1 z + a_2 z^2 + a_3 z^3 + ...$ where more than ...
1
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1answer
54 views

Define: Period & Fundamental Period

How period of a periodic function is different from its fundamental period? Distinction & similarity between period & fundamental period Authenticated definitions of period & fundamental ...
2
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2answers
48 views

Periodic continuous function which is integrable on $\mathbb{R}$

Let $f:\mathbb{R}\to\mathbb{R}$ be a $T$-periodic function, that is $f(t+T)=f(t)$ for all $t\in \mathbb{R}$. Assume that $$\int_0^{+\infty}|f(s)|ds<+\infty.$$ Now if we assume in addition that ...
0
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2answers
47 views

Period of $\frac{\sin(Ny)}{sin y}$ with $N$ odd?

The function $$f(y) = \displaystyle \frac{\sin(Ny)}{\sin y}$$ is periodic with period $2 \pi$ in general. But tracing the graphic of that function for $N$ odd it seems that for $0 \leq x < \pi$ ...
2
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0answers
25 views

Cayley on “trivial transformations”

In his 1854 paper, "Deuxième mémoire sur les fonctions doublement périodiques" ("Second memoir on doubly periodic functions"), Cayley discusses (what we would today describe as) a certain class of ...
2
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1answer
63 views

A calculus problem

Question: Suppose that $u(x,t)$ is continuous, together with its first and second partial derivatives; suppose that $u$ and its first partial derivatives are periodic in $x$ of period $1,$ and ...