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

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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 ...
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2answers
52 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 ...
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1answer
24 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. ...
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2answers
137 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
41 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 ...
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1answer
25 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 ...
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1answer
69 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 ...
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2answers
96 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.
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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 ...
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1answer
65 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 ...
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2answers
38 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
17 views
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21 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
26 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 ...
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2answers
36 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 ...
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0answers
27 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 ...
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1answer
30 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 ...
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2answers
46 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 ...
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2answers
46 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 $0 \leq x < \pi$ ...
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0answers
22 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 ...
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1answer
61 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 ...
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1answer
42 views

How to find out the period of fractional part of x

I came across this solved example in a book, it says - Find the period of the function : $f(x)=\sin(4\pi x)+\{3x\}$, where $\{x\}$ denotes the fractional part of $x$. Now I know that if $f(x)$ is ...
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3answers
113 views

A system of $n$ equations , how does it behave for growing $n$?

I read about the system of $n$ equations in the link below. I wonder how it behaves for growing $n$. Does it converge ? http://math.eretrandre.org/tetrationforum/showthread.php?tid=889 Here it is ...
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1answer
224 views

Real-analytic periodic $f(z)$ that has more than 50 % of the derivatives positive?

Im looking for a real-analytic function $f(z)$ such that for any $z$ $1) $$f(z+p) =f(z)$ With $p$ a nonzero real number and where $z$ is close to , or onto the real line such that $z$ is in the ...
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0answers
105 views

Is there a relationship between a function's period and number of roots?

Let: $f(x,a,l)=\prod _{k=a}^l\sin \left(\frac{\pi x}{k}\right)$ and $f(x,k)=\sin \left(\frac{\pi x}{k}\right)$ I came up with this equation to find the period $T(f(x,a,l))=2\,{{\pi }^{l-a+1} ...
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1answer
32 views

Promise of the hidden subgroup problem for $\mathbb{Z} mod 2$

I am going through the talk, Graph isomorphism, the hidden subgroup problem and identifying quantum states, by Pranab Sen. On slide 3, the promise of the problem is defined as follows. Here is the ...
3
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1answer
115 views

Approximate Periodic Function by shifting Basis Functions

Given a periodic "Target Function" $F(t)$ a set of $N$ periodic "Basis Functions" $B_i(t)$ of arbitrary shape All functions are defined on the same interval $T$. I am allowed to shift ...
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30 views

Period of hypocycloid

Hypocycloid is defined by the following parametric equations: $$x(\phi) = (a - b) \cos(\phi) + b \cos\big(\dfrac {a - b} b \phi\big)\\ y(\phi) = (a - b) \sin(\phi) - b \sin\big(\dfrac {a - b} b ...
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1answer
40 views

Hidden subgroup problem for $\mathbb{Z} mod 2$

The definition of the Hidden Subgroup Problem (HSP) is as follows (according to a lecture series by Pranab Sen), Let $G$ be a group, $S$ a set and $f : G \to S$ a function. We are given an ...
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2answers
52 views

Periodicity of a triginometric function

I have a trigonometric function and I'm interested to know whether or not it has a period. At this stage I'm fairly certain that it is not periodic. However, I don't know how to prove it. Can anyone ...
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1answer
34 views

Periodic expansion of function

Let $ \tilde{f}: \mathbb{R} \rightarrow \mathbb{R} $ be the 2-periodic expansion of the function $ f: [-1,1[ \rightarrow \mathbb{R} $ given by $ f(x)=x $. But how can I make a graph of $ \tilde{f} $ ...
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0answers
38 views

If $f$ is $2 \pi$ periodic and $\int_{0}^{2 \pi} f(t) dt=0$ then $\int_{0}^{2 \pi} (f(t))^2 dt \le \int_{0}^{2 \pi} (f'(t))^2 dt$ [duplicate]

Given $f$ a real differentiable function, $2 \pi$ periodic such that $\int_{0}^{2 \pi} f(t) dt=0$ show that: $\int_{0}^{2 \pi} (f(t))^2 dt \le \int_{0}^{2 \pi} (f'(t))^2 dt$. When does equality hold? ...
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1answer
38 views

Proving that a function is periodic

I need help proving the following: Let $f(x)$ be an even function and let $A$ be an arbitrary real number . If the function $g(x) = f(A - x) $ is odd then $f(x)$ is periodic.
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2answers
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Questions aobut Weierstrass's elliptic functions

from the wikipedia: == In terms of the two periods, Weierstrass's elliptic function is an elliptic function with periods ω1 and ω2 defined as $$\wp(z;\omega_1,\omega_2)=\frac{1}{z^2}+ \sum_{n^2+m^2 ...
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1answer
45 views

How to obtain a periodic function from a rapidly decaying function?

Suppose $f(x) = \exp(-x^2)$ with $x \in [0, 3]$. How could I periodise this function to obtain an analytical form of a continuum periodic function $x \in [0, +\infty)$ with period T = 3?
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103 views

Demonstration with Laplace

How can I demonstrate this? If $F(t)$ is a periodic function with a period of $T>0$, then $$ \mathcal{L}\{F(t)\} = \frac{\int\limits_0^T e^{-st} F(t)\operatorname d\!t}{1-e^{-sT}}\operatorname ...
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2answers
79 views

Non uniform continuity of a function and almost periodicity

We say that a continuous function $f:\mathbb{R}\to\mathbb{C}$ is almost periodic in the sense of Bohr if: For every sequence $(t'_n)_{n\geq0}$, there's a sub-sequence $(t_n)_{n\geq0}$ such that ...
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0answers
14 views

Phase reference of a periodic signal

Assume an arbitrary (discrete) signal that is periodic and known over a whole period. I need a way to select a characteristic point along the signal such that I can always retrieve it even when the ...
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1answer
77 views

$x'=Ax$ has one periodic solution. Prove that all solutions are periodic.

I want to prove the following: 1) Suppose $$A_{2,2}=\begin{pmatrix} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end{pmatrix}$$ real and suppose the system of differential equations ...
3
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1answer
64 views

What is this frequency mathematically? [closed]

I wonder what the graphing of these frequency is all about? Is it something in nature? See
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1answer
69 views

Relationship between Simple Harmonic Motion Equation and Wave Equation

I am very familiar with the equation: $$f(t)=A\sin(\omega t+\phi)$$ Used to describe the instantaneous value $f(t)$ of a wave with amplitude $A$, frequency $\omega$, and phase shift $\phi$ at time ...
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57 views

Periodic solutions to Riccati equations

Suppose $\alpha, L>0.$ Under what conditions (between $\alpha, L$) the Riccati equation $d\Phi/dz=2i[\Phi(z)^2+\alpha Cos(2\pi z/L)\Phi(z)+1]$ can have a periodic solution with period $L$ (under ...
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Mean Value of a Non Linear Function

If I have the following function $f(x)=1-x^c$ with $x$ being periodic and can be written in the form of $x(t)=Asin(wt)+B$. $c$ is real (most of the time not an integer) and $B$ is chosen in a way ...
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72 views

Periodic Solution of Riccati Equation

I want to know for which condition on "$a$" and "$k$", i.e. for which function of $a(k)$, the following Riccati equation, with the initial condition $u(0)=ia$ ($i^2=-1$), have periodic solution with ...
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1answer
48 views

Finding the period of complex exponential function

I am having some trouble finding the period of the following discrete signal: $x[n]=e^{jn2\pi/3}+e^{jn3\pi/4}$
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1answer
37 views

If $f$ is has period $\omega$ and a pole at $z_0$, prove that $f$ has a pole at $z_0+\omega$.

How do I show if $f$ is a meromorphic function with period $\omega$ and a pole at $z_0$, then $z_0+\omega$ is also a pole of $f$ with the same multiplicity and residue?
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2answers
209 views

Sturm Liouville with periodic boundary conditions

Background and motivation: I'm given the boundary value problem: $$y''(x)+2y(x)=-f(x)$$ subject $y(0)=y(2\pi)$ and $y \, '(0)=y \, '(2\pi)$. EDIT: These were not given to be zero !! Maybe this ...
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2answers
121 views

Are these statements of my professor about periodicity of harmonic processes in time series analysis correct?

Assume $X_t$ is a harmonic stochastic process, i.e., $$X_t = \sum_{j=-k}^k A_j \exp(i \lambda_j t)$$ where the frequencies $\lambda_j$ are given and $A_j$ are uncorrelated random variables with zero ...
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3answers
97 views

Period of $f(2x+3)+f(2x+7)=2$

I have a problem in finding the period of functions given in the form of functional equations. Q. If $f(x)$ is periodic with period $t$ such that $f(2x+3)+f(2x+7)=2$. Find $t$. ($x\in \mathbb R$) ...
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1answer
21 views

Complex Eigenvalue Periodic

This is probably a simple question, but I'm having trouble finding a clear answer. Let's say we have a system, with the complex eigenvalue: $$\lambda = \alpha + i\beta$$ I know that $\alpha < 0$ ...