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

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1answer
37 views

Integral of periodic function.

I was wondering if anyone can help me with this, if f(x) is a periodic function with period T then it satisfies $$\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)dx$$ for all $a \in \Bbb R$. It is clear that ...
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0answers
34 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
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1answer
22 views

How to calculate power of a non-continuous signal

I have to find the power of the following signal and would like to know if I'm doing this right or, if I'm doing it wrong, how to do it. The equation for power in my textbook is $\overline{m^2(t)} = ...
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0answers
23 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 ...
2
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1answer
25 views

How to find periodic points non-algebraically

For example, $f(x) = x - x^2$ by observation has a fixed point at $x = 0$, and an eventually periodic point at $x = 1$ (that goes to $0$). For the second iteration, to see if there's a periodic point ...
2
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1answer
44 views

Same Expansion for Different Functions! What's Wrong?

We know, the function $arctan(x)$ is defined as the real number $y$ such that $tan(y)=x$ and $-\frac{\pi}{2}\leq y\leq\frac{\pi}{2}$ From this definition we can derive the well-known ...
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9 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 ...
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1answer
35 views

It's possible to prove that this integral is positive?

I would like to prove that this function $$F_t(x)=\int_{-\infty}^{\infty}e^{-\frac{u^2}{2 t}}\,\cosh\left(u\right)\,K_{1/2+i\,u}\left(\frac{1}{4x}\right)\,du,\hspace{0.5cm}x,t>0$$ is positive, ...
1
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1answer
48 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 $T$, ...
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2answers
13 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 ...
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1answer
33 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 ...
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1answer
19 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
17 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
57 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 ...
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0answers
23 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
31 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
67 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 ...
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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 ...
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1answer
27 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 $f$. ...
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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 ...
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2answers
36 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.
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3answers
123 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 ...
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2answers
33 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
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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?
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1answer
20 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 ...
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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
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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 $n^2$?
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1answer
52 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...?
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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
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1answer
59 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
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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 ...
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0answers
17 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 $[-L,L]$. ...
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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. ...
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2answers
83 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}} ...
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1answer
37 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 ...
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1answer
59 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 ...
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2answers
88 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 ...
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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
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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 ...
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1answer
54 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 ...
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0answers
42 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 $-\Delta u = f$ ...
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3answers
65 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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21 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 ...
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3answers
47 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
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1answer
65 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 ...
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3answers
55 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
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1answer
62 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
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2answers
74 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 $0$ ...