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

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0
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1answer
30 views

The period of a non-linear pendulum

The period of a non-linear pendulum is $T = \sqrt{2} \cdot \int_{-\theta_0}^{\theta_0} \frac{d{\theta}}{\sqrt{\cos(\theta) - \cos(\theta_0)}}$. My problem: what will happened with the period $T$, ...
0
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0answers
18 views

Period of a non-linear pendulum

I have proved that the period of a non-linear pendulum is $T(\theta_0) = \sqrt{2} \cdot \int_{-\theta_0}^{\theta_0} \dfrac{d{\theta}}{\sqrt{\cos(\theta) - \cos(\theta_0)}}$. I need to show, that ...
0
votes
2answers
10 views

Given two piece-wise functions, prove their composition is periodic.

I know how to graph h(f(x)), however, I am having trouble proving it is periodic. i. We must prove that h(f(x + k)) = h(f(x)) for all x ∈ ℝ and for some k ≠0 ∈ ℝ in order to prove that h(f(x)) is ...
0
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1answer
28 views

Period of a function?

I am trying to find out the period of a function but this function is giving me a different answer from what I expected . f(x) = |sinx| + |cosx| I know that to find the period of sin and cos we ...
-1
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1answer
18 views

Complex Fourier Series of $t^3$

I am trying to find compute the complex Fourier series of the following function: $$f(t) = t^3$$ $$-\frac32 \le t \le \frac32$$ $$f(t) = f(t+3)$$ I am using the generic function for the complex ...
0
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1answer
22 views

Show that $n\bmod m$ is periodic $\forall n,m \in \mathbb{N}^+$

How can I show that $n\bmod m$ is periodic? If I have a simple example like $n\bmod6 \equiv a$ how can I show that this is periodic if e.g $f(n) = n\bmod6$?
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0answers
16 views

How transform an oscillatory integral into a non-oscillatory integral

I would like to ask if anybody knows how to change an oscillatory integrand into a non-oscillatory integrand. That is $$\,\,\int_a^b ...
1
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1answer
55 views

Evaluating an integral of a periodic function

I have been several days trying to prove that the following integral $$I_t(x)=\int_{-\infty}^{\infty}e^{-\frac{\cosh^2(u)}{2x}}\,e^{-\frac{u^2}{2 t}}\,\cos\left(\frac{\pi\,u}{2t ...
-1
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0answers
20 views

Create periodic function from points

Im doing a some small research and i come up with the need of creating a periodic function from the given values: 7, 49, 77 , 91 , 119, 133, 161 , 203 , 217 From 217 the pattern repits, which means ...
2
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1answer
30 views

On the weak closedness of a closed ball with fixed $L^2$-norm in a periodic Sobolev space

Preliminaries: Let $\mathrm{L}_P^2$ denote the Hilbert space of $P$-periodic, locally square-integrable functions $f\colon \mathbb{R} \to \mathbb{C}$ with Fourier series representation $$f(x) \sim ...
2
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2answers
65 views

How to prove $ I_t=\int_{0}^{\infty}g(u)\,\cos\bigl(\frac{\pi\,u}{2t }\bigr)\,du$ is positive?

We have the following integral $$I_t=\int_{0}^{\infty}g(u)\,\cos\left(\frac{\pi\,u}{2t }\right)\,du\,\,\,\text{with}\,\,\,\,t>0$$ where $g(u)$ is continuous on $(0,\infty)$, $g(0)=1$ and $g(u)$ is ...
1
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0answers
16 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 ...
0
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0answers
23 views

If $f(t)$ is periodic, is there any $t$ that would equal to DC components?

Suppose $f(t)$ is periodic with period $T$. Would there be $t$ that would necessarily equal to DC component (it can be scaled)? By DC component, I mean $F(0)$ where $F$ is fourier coefficient of $f$. ...
0
votes
2answers
23 views

Simple harmonic motion~

I am stuck in a question which says that: A particle moves on the X- axis according to equation $x=A+Bsin(\omega t)$. The motion is simple harmonic. Find the amplitude of SHM. The answer of the above ...
2
votes
2answers
34 views

How do I find the period of the sine function $y = 20\sin\left[\frac{5 \pi}{2}\left(\frac{x -2}{5}\right)\right] + 100$

Using Desmos I can see the period is $0.8$ but how do I get there? I understand that the period is $2\pi/$co-efficient of $x$ but the $-2/5$ is throwing me off.
5
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3answers
118 views

Improper Integral of a periodic function converges

Given $f(x)$ is a periodic function and $\int_0^p{f(x)}dx=0$. Show $\int_1^\infty\frac{f(x)}{x}dx$ converges. 1) I know this integral can be broken into ...
1
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2answers
32 views

Periodic Functions f(x)=r(x)/s(x)

I have two questions in one. (1) Let $f(x)=\frac{s(x)}{r(x)}$. If $s(x)$ and $r(x)$ are polynomials of same degree, prove that if $f(x)$ is periodic, it must be constant for all ...
4
votes
1answer
63 views

Behavior of a periodic function

Can a periodic function satisfy $f''(x)f(x)>0, x\in \mathbb{R}$ My intuition says no. Any thoughts on how to approach this?
0
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1answer
18 views

Implication of period of derivative

Let $f:(0,1)\rightarrow\mathbb{R}$ be a function with nonconstant derivative $f'$ on $(0,1)$. Is it possible that there exists a real number $t$ such that $f'(x)=f'(x+t)$ for all $x$ such that ...
1
vote
1answer
22 views

prove that $f$ periodic, continuous, $f(x+r)=f(x)+c$ cannot be satisfied with $r\ne 0$ and $c\ne 0$

Prove that if $f$ is periodic, continuous, there are no $r\ne 0$ and $c\ne 0$ constants satisfying $f(x+r)=f(x)+c$ (for all $x$) Let's assume that exist $r$ and $c$ satisfying $f(x+r)=f(x)+c$. Thus, ...
1
vote
1answer
150 views

Frequency of a solution to a PDE

Why is the frequency of $u=\sum_{n=1}^{\infty}B_n$sin$(n\pi x/L)$cos($n^2 \pi^2ct/L)$ equal to the coefficent of $t$ with $L>0,c$ constants, i.e Why is it proportional to $n^2$?
0
votes
1answer
49 views

Can this be a periodic function

$$F(x) = \sqrt{\sin x + \cos x}$$ I graphed this function and it seemed to be periodic but my textbook says that it's not. Is any other rule it's breaking...?
1
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0answers
29 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 ...
2
votes
1answer
52 views

Solution of differential lyapunov equation

How would I solve for following, else any implemented algorithms or solvers in matlab (even ways to solve it) for Lyapunov differential equation of form: $$P'(t) + A(t)^TP(t) + P(t)A(t) + Q(t) = 0,$$ ...
0
votes
1answer
19 views

$U_{n+1} = a * U_n * (1-U_n)$ with : $3<a<3.6$ What are some values of $a$ such that $U_n$ changes its periodicity?

$U_{n+1} = a * U_n * (1-U_n) = a * U_n - a * (U_n)^2$ with : $3<a<3.6$ What are some values of $a$ such that $U_n$ changes its periodicity? I've computed $U_n$ for many different values of ...
0
votes
0answers
16 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
1answer
17 views

Periodicity of discrete function

Let $x[n]$ a discrete-time signal, $$y[n]= x[2n]$$ I have seen that if $x[n]$ is periodic then $y[n]$ is periodic. Similarly, can we say that if $y[n]$ is periodic then $x[n]$ is periodic as well. ...
0
votes
2answers
82 views

solve integral of $\frac{\sin (ax)}{ \sin(x)}$

I would like to find the area under the curve of $\frac{\sin(ax/2)}{\sin(x/2)}$, namely between the first zero crossing on the left and right: $$ \int_{-\frac{2\pi}{a}}^{\frac{2\pi}{a}} ...
0
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1answer
29 views

Modeling with periodic functions

In the month of March, the temperature at the South Pole varies over the day in a periodic way that can be modeled approximately by a trigonometric function. The highest temperature is about ...
1
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1answer
53 views

Showing Weierstrass Elliptic Function is meromorphic

Consider the Weierstrass Elliptic function, $$\rho(z) = \frac{1}{z^{2}} + \sum\bigg(\frac{1}{(z-m-nw)^{2}} - \frac{1}{(m+nw)^{2}}\bigg), $$ where $m,n \neq 0,0 $. How can one show that it is ...
1
vote
2answers
86 views

Solve $\sin(ax) / \sin(x) = a/2$ for $x$

I am currently trying to solve $\sin(ax)/\sin(x) = a/2$ for $x$, where $x$ is between $0$ and $\pi$ and $a$ is a constant. As I have limited skills in math, I cannot seem to solve this problem ...
0
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0answers
22 views

Proving a function elliptic.

Consider the elliptic function, $\rho(z) = \frac{1}{z^{2}} + \sum(\frac{1}{(z-m-nw)^{2}} - \frac{1}{(m+nw)^{2}}) $ here $m,n \neq 0,0 $. The task is to prove this function elliptic, so doubly periodic ...
4
votes
2answers
100 views

prove that $f$ is periodic

A function $f\colon\mathbb{R}\to(0,\infty)$ satisfies equation $f(x)=f(x+64)+f(x+1999)-f(x+2063)$. Prove that $f$ is periodic. I'm quite sure that 1999 and 64 are random numbers (probably 1999 = year ...
-1
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1answer
53 views

Show that $x \cos(cx)$ is aperiodic

I'm using the function $f(x)=x~cos(cx)$ in a paper and the periodicity of the function is relevant. Is there a simple way to show that a function of this form is aperiodic, or is it reasonable to ...
0
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0answers
35 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 $-\Delta u = f$ ...
1
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3answers
64 views

Different Types of Waves

I am making a basic 2D rigid body simulator as a hobby. It involves springs. Naturally, I need to render them. Rigid body simulators, such as Algodoo, render them simply like this Another (more ...
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0answers
20 views

Periodic Function With A Given Functional Equation

Let $f$ be a real valued function defined for all real numbers $x$ such that for some positive constant '$a$' the equation $f(x+a)=1/2+\sqrt{f(x)-(f(x))^2}$ holds for all $x$. Then prove that $f$ is ...
0
votes
3answers
46 views

Fourier Series - Periodicity

I don't know if I'm doing something wrong in this exercise. $f(t)=\pi-t$, if $0<t<\pi$ $f(t)=0$, if $\pi<t<2\pi$ I have to find the Fourier Series of $f(t)$ I define the Fourier ...
4
votes
1answer
64 views

Prove or disprove that $\sin\lfloor x\rfloor$ is periodic.

The title says it all. I was plotting random functions on my phone and noticed this graph. I don't think this function is periodic (WA also agrees). Is there a way to prove if a function like this is ...
0
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3answers
51 views

Find the period of $|\sin x| + |\cos x - 1|$

I want to find the period of this function , I know that the period of $|\sin x| + |\cos x|$ is $π/2$ but what can a $-1$ do?
4
votes
1answer
60 views

What is the period of $\sin 2\theta + \sin \frac{\theta}{2}$ [duplicate]

What is the period of $\sin 2\theta + \sin \frac{\theta}{2}$? The period of the first term is $\pi$ and that of the second is $4\pi$. Does that mean that the period of the whole is $4\pi$?
0
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1answer
33 views

Confused about Fourier series?

From linear algebra we know that if a set of vectors form a basis for a space, their is a unique linear combination of the basis to form any vector in that space. I'm assuming this extends to scalar ...
3
votes
2answers
68 views

Constructing A Space Filling Curve that fills the Unit Square

I'm reading Neal Carothers' Real Analysis, and he constructs a curve defined over $[0,1]$ that fills the unit square as follows: Let $f$ be a real-valued function over $[0,1]$ such that $f$ is $0$ ...
2
votes
0answers
49 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 ...
0
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1answer
42 views

Derivates of periodic parametric cubic splines

My Problem is sort of solved, I overlooked, that paameters $B$ to $D$ are dependent on $x$ and $y$ one question remains, see bottom of question. I implemented a periodic parametric cubic spline, and ...
0
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1answer
42 views

Periodic functions proof

I need some help here. Let $f$ be a $2\pi$-periodic function, and define for an arbitrary $k\in\mathbb N$ a function $g(x) = f(kx)$. Show that $g$ is also $2\pi$-periodic. What I've done: $$ g(x) ...
4
votes
1answer
54 views

Fourier coefficients intuition?

I just learned about Fourier series, and this is how I interpreted them: The complex exponentials form a basis for all periodic functions, and the Fourier series essentially decompose the function ...
3
votes
1answer
21 views

Is this function/series periodic?

$$f(t)=\sum_{k=-\infty}^{\infty}(-1)^kp_{0.5}(t-2k)$$ Recall: $$p_{\Delta}=\begin{cases}\frac{1}{\Delta},&0\leq t\leq\Delta\\0&\text{ otherwise.}\end{cases}$$ Is the function periodic? If ...
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, ...
0
votes
1answer
51 views

Dilating sinusoidal period at certain abscissa $x$?

Lets take a look at this function : $$f(x) = \sin\left(\frac{\pi x }{2}\right)$$ when $x$ tends to $1$ this functions get closer to $1$ by bigger values now look at this one : ...