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

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8
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141 views

number of zeros of complex waves

Does anybody know about any type of methods how to calucalte/estimate the number of the zeros of complex waves (periodic functions as superposition of many harmonic waves) within a given period [0,x] ...
3
votes
0answers
54 views

Is there exact formula that returns minimal period of a periodic function?

Is there exact formula that returns minimal period of a periodic analytic function? For constant it should return 0, for non-periodic functions - infinity. I only came to the following but it ...
3
votes
0answers
81 views

Differential Equation has a unique solution periodic

Let $A(t)$ continuous and periodic of period $S$ in $\mathbb{R}$. Suppose $x' = Ax$ has $\varphi \equiv 0$ as the only periodic solution of period $S$. Show that there exists $\delta> 0$ such that ...
3
votes
0answers
128 views

Computing the Fourier transform (in the sense of tempered distributions) from a sequence of samples

Let $f\in L_{\operatorname{loc}}^2 (\mathbb{R})$ be a function of period $2\pi$. I'd like to get some help proving the following identity: $$\hat{f}=\sum_{n=-\infty}^{\infty} \hat{f}(n)\delta_n$$ ...
3
votes
0answers
578 views

Integration of nontrivial trigonometric functions

First an example which I know how to solve. If we have the following integral $$\int_{-\pi}^{\pi}\frac{1}{1+3~\cos^2(t)}dt$$ there is a very practical way to evaluate it by interpreting it as some ...
2
votes
0answers
23 views

Find the fundamental period

How do I find the fundamental period of this function? $$y = \sin x + \cos(1,01x)$$ I know that the fundamental period of $\sin x$ is $2\pi$ and the fundamental period of $cos(1,01x)$ is ...
2
votes
0answers
25 views

Is it Possible to represent $f(x) =\arctan(x)$ as a fourier series ? Why?

Is it Possible to represent $f(x) =\arctan(x)$ as a fourier series ? Why ?
2
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0answers
17 views

Riemann Integrals and Periodic Functions

Consider the following result, where integrals are, say, Riemann integrals: Let $f:\mathbb{R}\to\mathbb{C}$ be a periodic function of period $L$. Then for all $a\in\mathbb{R}$ $$ ...
2
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30 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
0answers
27 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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86 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 ...
2
votes
0answers
116 views

Periodic Laplace transform

Here's the questions and the graph I've been struggling with this since Thursday and this is due today. I need help on problems a and b. For a, the question is: "Write $f(t) = \sum_{n=0}^\infty ...
2
votes
0answers
57 views

Does weak convergence (of a periodic function) imply weak-* convergence for the derivative?

Lets assume we have $b(x,t) \in L^\infty$ periodic in x and $\frac{db}{dt} \in L^{p}, p\in\mathbb{N}$. $b(x,t)$ converges weakly to an $f(t)=\int b(x,t)dx \in L^q$ for all $1<q<\infty$. ...
2
votes
0answers
58 views

Do there exist periodic fractals $A_f$ of this type?

Let $z$ be a complex number. Meromorphic here means meromorphic on all of the complex plane $C$. Lets define a fractal $A_f$ on the complex plane as the result of iterating a meromorphic function ...
2
votes
0answers
250 views

Fourier Series problem

Suppose you are given the following information about a continuous-time periodic signal, $x(t)$, with period $6$ and its Fourier series coefficients $(a_k)$, (1)-(4). Using the synthesis equation, ...
1
vote
0answers
20 views

period of the sum and the product of sin(ax) and cos(bx)

Take the function f(x)=sin(ax)cos(bx) , with a,b>0 . We know that the fundamental period of sin(ax) is p= 2π/a and the fundamental period of cos(bx) is q=2π/b. Suppose there are positive integer n ...
1
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0answers
26 views

Ode with Piecewise function

We can write this $$12x"+36x'+48x=f(t)$$ my main problem is how to solve this non-homogeneous ODE I know how to do this as 2 different ode unfortunately its not in a syllabus which doesn't use ...
1
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0answers
26 views

integral of periodic function with interval to integrate outside definition of function

So the task is to integrate $${\frac{1}T}\int_0^{2\pi}tf(t)\,dt$$ where $T=2\pi$ and $$f(t) = \left\{ \begin{array}{ll} -t^2 & \quad -\pi < t < 0, \\ t^2 ...
1
vote
0answers
78 views

Car traveling on a bumpy road (ODE)

The suspension system of a car traveling on a bumpy road has a stiffness of $k = 5\times 10^6$ N/m and the effective mass of the car on the suspension is $m = 750$ kg. The road bumps can be considered ...
1
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0answers
23 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 ...
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0answers
33 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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0answers
106 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} ...
1
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0answers
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 ...
1
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0answers
61 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 ...
1
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0answers
62 views

Definite Integral of Periodic Function Multiplied by another Function

For one part of a problem I am working on, I am trying to show that $y'(t) \geq 1$ for all $t \geq 0$ when $y'= 1- \int^t_0 g(s)y(s) ds$ When $g(t)$ is periodic, $g(t) <0$ for all $t$, and ...
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49 views

Help with the system of 13 equations

Homework from electronics class, but since it looks like a math problem to me i decided to look for help here. I am given this composite periodic signal that has 6 harmonics: ...
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27 views

Solutions for ODE with periodic B.C.

I have the following ODE $$-u(x)''=f(x) \qquad u:[0,1)$$ It is smooth and 1-periodic. Assume that I have a solution u(x). How do I prove that: $$u(x)= u(x)+c \quad c \in \mathbb{R} $$ is also a ...
1
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0answers
58 views

Triangulation in periodically repeated 3D box

I have a set of points in a 3D box (rectangular parallelepiped), which is periodically repeated out of two its opposite sides. How do I find the Delaunay triangulation for this set of points? Here is ...
1
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31 views

Fourier transform of $f[n]$ and $f[-n]$

Hi I am just wondering, If I have a signal $f[n]\in \mathbb{C}^L$, i.e. $f$ is $L$-periodic, i can also define $h[n]=f[-n]$. Is it true that the Fourier transform of $f$, say $\hat{F}$, and the ...
1
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0answers
84 views

Is there a generic approach to Generating Function of periodic sequences?

Recently I read on wiki (see here): "Any periodic sequence can be constructed by element-wise addition, subtraction, multiplication and division of periodic sequences consisting of zeros and ones." ...
1
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0answers
555 views

Solving a Forced Oscillations Differential Equation Problem

A building consists of two floors. The fi rst floor is attached rigidly to the ground, and the second floor is of mass m = 1000 slugs (fps units) and weighs 16 tons (32,000 lb). The elastic frame of ...
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0answers
112 views

Inequalities of integrals of periodic functions

I have a function that has a shape similar to $\sin(x)^2$ (could be periodic extensions of $(x/(\pi/2))^2$ defined between $-\pi/2$ to $\pi/2$ for example). Let's call it $g(x)$. I want to show that ...
1
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0answers
109 views

sum of periodic function which eventually vanish

It is known that : If $\lim_{x \to \infty} \sum_{i=1}^n f_i(x)=0$ and $f_i$ are real-valued periodic function, then $\sum_{i=1}^n f_i(x)=0$ for all $x \in \mathbb{R}$. We can solve this problem by ...
1
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0answers
32 views

Asymptotic order of some sums with the Fourier coefficients

Given $f\in C^{w}[0,1]$ with periodic conditions $f(0)^{(j)}=f(1)^{(j)},\ j=0,\dots, w-1$ and its Fourier series are $f(x)=\sum_{l}f_{i}\exp(2\pi ix)$. I need to find the asymptotic order of errors ...
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0answers
123 views

Non-periodic BV function

I want to know the definition of non-periodic bounded variation function. I know the definition for periodic function of bounded variation, which is, Let $f:[a,b]\to \mathcal c$ and ...
0
votes
0answers
36 views

Properties of periodic solutions of nonlinear ODE system

Assume you have a complicated nonlinear ODE system with some parameter $p$. Numerical simulations of the system show, that for any initial conditions and $p$, the solution tends to a periodic function ...
0
votes
0answers
11 views

Order of distribution of the zeros of the interference function of periodic oscillations?

Given a finite (or an infinite) number of periodic oscillations of different shapes but even functions, along the abscissa $x∈R$, every such periodic oscialltion may cross on some zero points the ...
0
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0answers
20 views

Extremum points of Periodic functions

Assume that the function $f,g:\mathbb{R}^n\to\mathbb R$ are $2\pi$-periodic continuous functions (so continuous functions defined on torus $\mathbb T^n$), differentiable almost everywhere in $\mathbb ...
0
votes
0answers
34 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 ...
0
votes
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 ...
0
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0answers
20 views
0
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0answers
16 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 ...
0
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0answers
19 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 ...
0
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0answers
23 views

Linear Differential Equation with Quasiperiodic Coefficient: A Question

I am reading the book Synergetics. The following passage is from the book: Assume that in the differential equation $\dot{q}= a(t)q$, where $a(t)$ is quasiperiodic, i. e., we assume that $a(t)$ ...
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0answers
9 views

analytic property of periodic properties

Can any one suggest me some books in which I can see the analysis of periodic functions? I dont have any constrain in the domain or codomain. For example this book ...
0
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0answers
25 views

Mathematical project on ocean waves and buoyancy

I am currently searching for an idea for a math project on actual ocean waves and water volume displacement. I am thinking of creating a general formula or ratio of crest to trough volume for when a ...
0
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0answers
59 views

Explicit formula for inclusion/exclusion

I've been searching for a formula for the cardinality of the union of $n$ sets but all the formulas I can find incorporate the symbol (...) and summations that have limits of the form ...
0
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0answers
31 views

problem about Period

we have a, b, m initially where 0 <= a, b < m <= 10^7. And we define a[1] = a, ...
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447 views

Find the period of sequence.

A sequence is such that its terms are generated by the formula: $$r_i =(ar_{i-1}+b\pmod m)$$ where $a,b,m,r_0$ are given. find the period,that is the number of terms that are repeated. For example, ...
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69 views

Factoring from order finding?

It appears that if you have a finite field, F_p, and you can factor the order of the field, p-1, then you can easily construct an arbitrary nth root of unity in F_p and also check that it be primitive ...