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

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7
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0answers
155 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] ...
4
votes
0answers
45 views

Finding period of a function? Am I doing something wrong?

I need to find the period of the function $$\large{\frac{\sin (\sin {nx})}{\tan{\frac{x}{n}}}}$$. According to me, the period of $\sin (\sin (nx))$ should be $\large{\frac{2\pi}{n}}$ and the period ...
3
votes
0answers
74 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
85 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
764 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
22 views

Solving differential equation with Fourier-series-inhomogenity

Let $\lambda$ be a real number , $(c_k)$ a complex sequence with $\mid c_k \mid \leq C(1+\mid k \mid)^{-2}$ for all k with a constant $C \geq 0 $. Find all periodic, two times differentiable ...
2
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0answers
57 views

Periodicity of a sum of periodic functions?

The sum of two periodic functions is periodic if: a) Both periodic functions are continuous b) If the ratio of their fundamental periods is rational Can someone explain why the first ...
2
votes
0answers
39 views

When does a linear map become the identify map?

My question is about a linear map defined on the set of smooth periodic functions. Precisely, let $C$ be the set of infinitely many times continuously differentiable and $2\pi$ periodic functions, ...
2
votes
0answers
73 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
118 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
votes
0answers
31 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
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
0answers
31 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
0answers
105 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
122 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
69 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
69 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
341 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
16 views

Why is $\int\limits_{u}^{u+\omega_j} f'(z)/f(z) dz \in 2 \pi i \mathbb{Z}$?

Because $f(u) = f(u+\omega_j)$ for $j \in \{1,2\}$ it applies $$\int\limits_{u}^{u+\omega_j}\frac{f'(z)}{f(z)}dz \quad \in 2\pi i \mathbb{Z} \quad \text{for} \quad j=1,2, $$ Hello, I write my thesis ...
1
vote
0answers
17 views

How to check periodicity of $f(t)$ using samples

Suppose that we know that signal $f(t)$ is $T_1$-periodic. Let $f_1 = 1/T_1$. But we want to know whether signal is $T_2$-periodic also. Let $f_2 = 1/T_2$, and $f_2$ is positive integer multiples of ...
1
vote
0answers
39 views

Existence of periodic orbits (non-linear systems)

I'm trying to solve the following problem: Use the Poincaré-Bendixson's criterion to show that the system has a periodic orbit $$ \dot{x}_1 =x_2 \\ \dot{x}_2=-x_1+x_2-2(x_1+2x_2)x_2^2 $$ The unique ...
1
vote
0answers
36 views

Find what values of 'b' have bounded solution(differential equation)?

$y′′ + b^2{y} = f(t)$ $ f(t) = t$ for $0 < t < 2\pi$ ($2\pi$ periodic sawtooth wave) This is my solution to the differential equation. $y(t) = C_1\cos(bt) + C_2\sin(bt) + b^{-3}\left(bt - ...
1
vote
0answers
146 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
vote
0answers
30 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
vote
0answers
39 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
155 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
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
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} ...
1
vote
0answers
37 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
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 ...
1
vote
0answers
78 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 ...
1
vote
0answers
50 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: ...
1
vote
0answers
32 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
vote
0answers
65 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
vote
0answers
102 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
vote
0answers
738 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 ...
1
vote
0answers
124 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
vote
0answers
115 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
vote
0answers
33 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 ...
0
votes
0answers
24 views

On Period of Linear Recurring Sequences modulo $P^e$

If a sequence $ X_0,X_1,X_2,\ldots$ is defined in terms of an initial set $ X_0,X_1,X_2,\ldots ,X_{k-1} $ by the recurrence relation $$ X_{n+k}= ...
0
votes
0answers
17 views

Scalar product with $f(x,y)$ when $ \dot{X} = f(X)$ has periodic orbit

Let $g(x,y), f(x,y) \in C^1: R^2 \to R^2$ such that $f(x,y) \cdot g(x,y)=0$, $\forall (x,y).$ Prove that if $\dot X = f(X)$ has a periodic orbit, then $g$ have a root Intuitively I can see that the ...
0
votes
0answers
25 views

about Poincare map

I saw that the Poincare map is defined by the flow of the periodic system with the least period $T$. that is, $$P(x):=\phi_T(x)$$ is a Poincare map with flow $\phi$ of time $T$. but I think if we ...
0
votes
0answers
30 views

Maximum period-length in decimal chain-addition?

Decimal chain-addition $^*$ takes a finite initial "seed" string of decimal digits, say $x_1x_2...x_n$, and defines an infinite periodic string $x_1x_2...x_n...$ by iterating the following rewrite ...
0
votes
0answers
36 views

Is this function $2\pi$-periodic?

If I construct $$G(x) =\sum_{1}^{\infty} g(x +2n\pi),$$ does this make $G(x)$ $2\pi$-periodic? My understanding is that if $G(x)$ were now $2\pi$-periodic, then that means $G(x) = G(x + 2\pi$) = G(x ...
0
votes
0answers
26 views

Show a function is periodic and find the period

Let $x(t)$ be a continuous signal, and $\hat x(u)$ be the fourier transform of $x(t)$. We define $\sigma_T(u)=\frac{1}{T}\sum_{n=-\infty}^{\infty}\hat x(u-\frac{n}{T})$ Show that $\sigma_T$ is ...
0
votes
0answers
15 views

What is the nomenclature for the repeating part of a curve with n-repeating-peaks?

Below is a Google Trends search for "past papers", notice the curve has repeating portions where each repeat has three peaks at different levels. I want to know what the technical name of such a ...
0
votes
0answers
25 views

How to write a periodic function expression from a piece of another function

How can I write a periodic function expression from a piece of another defined function?, e.g., how to write a periodic function using the piece of the function $f(x) = x^2$ in the interval $[-L,L]$. ...
0
votes
0answers
59 views

Laplace problem with elliptic BC - FINITE DIFFERENCES

I am experiencing some trouble in solving the problem $u-\Delta u = f$ with periodic boundary conditions. First question: Is the problem always solvable? If I am not mistaken $-\Delta u = f$ ...
0
votes
0answers
34 views

Finding the period of the function

My question is as follows:- Find the trigonometric interpolant $ \bar{f}(x)$ for $f(x)= \frac{\pi}{x+3\pi}$ and $n=1$. Thant is, find coefficients $c_{-1}, c_0 ,c_1$ such that ...
0
votes
0answers
69 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 ...