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

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11
votes
2answers
264 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
votes
1answer
104 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 ...
1
vote
2answers
28 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
votes
1answer
44 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
vote
4answers
435 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
vote
2answers
266 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
votes
1answer
41 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
votes
2answers
249 views

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

$\sin\sqrt x$ is not a periodic function. How can one prove this?
0
votes
2answers
302 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
vote
1answer
149 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
votes
1answer
101 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
votes
2answers
108 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
votes
1answer
35 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
votes
1answer
781 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
vote
2answers
125 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?
0
votes
2answers
32 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 $\frac {T} {2}$ ...
1
vote
0answers
43 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
vote
2answers
469 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
votes
0answers
32 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 ...
2
votes
2answers
672 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
votes
2answers
61 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
votes
2answers
56 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
votes
0answers
29 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
votes
1answer
70 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 ...
1
vote
1answer
177 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 ...
1
vote
3answers
120 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 ...
8
votes
1answer
237 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 ...
1
vote
0answers
109 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} ...
3
votes
1answer
42 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
votes
1answer
158 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 ...
1
vote
0answers
36 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 ...
3
votes
1answer
47 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 ...
1
vote
2answers
62 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 ...
0
votes
1answer
40 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} $ ...
5
votes
0answers
46 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? ...
0
votes
1answer
47 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.
4
votes
2answers
416 views

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 ...
0
votes
1answer
63 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?
0
votes
2answers
122 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 ...
2
votes
2answers
108 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 ...
0
votes
0answers
21 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 ...
5
votes
1answer
147 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 ...
0
votes
1answer
452 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 ...
1
vote
0answers
65 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 ...
0
votes
0answers
22 views

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 ...
2
votes
0answers
103 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 ...
0
votes
1answer
1k views

Finding the period of complex exponential function

I am having some trouble finding the period of the following discrete signal: $x[n]=e^{jn2\pi/3}+e^{jn3\pi/4}$
1
vote
1answer
40 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?
7
votes
2answers
392 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 ...
5
votes
2answers
142 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 ...