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
158 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
65 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
37 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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2answers
94 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: $$P'(t) + A(t)^TP(t) + P(t)A(t) + Q(t) = 0,$$ ...
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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 ...
0
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0answers
22 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
18 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
86 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
62 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
67 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
91 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 ...
1
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1answer
41 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 ...
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2answers
102 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
55 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
66 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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1answer
33 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
64 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
66 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
64 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
67 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 $4\pi$?
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1answer
35 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 ...
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2answers
79 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 $0$ ...
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0answers
56 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
48 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
65 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
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1answer
56 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
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1answer
22 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
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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, ...
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1answer
56 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 : ...
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0answers
32 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 ...
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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 - ...
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2answers
42 views

Frequency of sinusoidal curve

In this site,The frequency of a trigonometric function is defined as the number of cycles it completes in a given interval. The formula is : frequency=1/period The period of a sine function is ...
1
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1answer
88 views

Periodic solution to $y\prime = ay + b(x)$

I'm a bit stuck on a certain review question for ODEs. Here's how it goes: Given the one dimensional equation $y\prime(x) = ay+b(x)$, with $a\neq 0$ and $b:\mathbb R\to \mathbb R$ continuous and ...
5
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1answer
164 views

Prove that a $1$-periodic function $\phi$ is constant if $\phi(\frac{x}{2}) \phi(\frac{x+1}{2})$ is a constant multiple of $\phi$

For $0 < x< \infty$, let $\phi (x)$ be positive and continuously twice differentiable satisfying: (a) $\phi (x+1) = \phi (x)$ (b) $\phi(\frac{x}{2}) \phi(\frac{x+1}{2}) = d\phi(x),$ where ...
3
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3answers
137 views

Understanding the periodicity of a complex exponential function

In the reals, $e^{nx}$ explodes to infinity very fast. But, $e^{inx}$ is bounded and periodic. I am familiar with Euler's formula $e^{ix} = \cos x +i\sin x $. Yet, could you give me some intuition ...
1
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1answer
39 views

Why is a wave with high FM aperiodic?

I was playing with sound synthesis in a program I wrote and I had a wave of the form $\sin(2\cdot\pi\cdot(f_c+\sin(2\cdot\pi\cdot f_m \cdot t)) \cdot t) $ So, just simple frequency modulation. When ...
1
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1answer
52 views

how can we show an antiperiodic function?

How can we graphically show an anti-periodic function? I can't imagine. Maybe I have got no imagination...!! for example we can show the sin and cos or other periodic functions on the graphs. is it ...
2
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0answers
72 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 ...
4
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1answer
65 views

How many distinct values of floor(N/i) exists for i=1 to N.

Say we have a function $F(i)=\text{floor}(N/i)$. Then how many distinct values of $F(i)$ will exist for all $0 \leq i \leq N$ e.g. We have $N=25$ then. $F(1)=25$ $F(2)=12$ $F(3)=8$ $F(4)=6$ ...
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1answer
59 views

How to determine general solution using Laplace transform?

$y′′ + a^2y = 2u(t-10)$ Here $a > 0$ and is any real number. I am confused by $a^2$ value there. Can anyone show me step by step how to get to up to $Y(s)$? It would really help me understand. ...
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2answers
103 views

How to solve differential equation problem involving Dirac delta function?

$$ y''+2y'+ 10y=b\,δ\left(\, t - T\,\right)\,\qquad y\left(\, 0\,\right)=3\,,\quad y'\left(\, 0\,\right)=0 $$ Can you choose values for $b$ and $T$ ( $b$ and $T$ positive numbers) such that ...
1
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1answer
41 views

Differential equation, for which values of 'a' does this have a bounded solution?

Let $f(t) = f(t$) be the 2pi periodic("sawtooth wave"), f(t) = t for $0 \leq t \leq 2\pi$ and consider the equation $$y^{\prime \prime} + a^2y = f$$ For which values of $a$ (here $a$ >0) does this ...
2
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1answer
43 views

How to find solution of the integral equation?

$$y(t) + t \int_0^t y(v)dv = 1 + \int_0^t vy(v)dv$$ I found the answer to be $y(t) = \cos{t}$. I have no idea how they go this answer. I would appreciate any suggestions how to solve this.
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1answer
47 views

A formal justification for this “physicism”?

I gave a presentation for a seminar class yesterday on Fourier analysis, and introduced the sawtooth function as a counterexample, for a function whose Fourier series is not termwise differentiable. ...
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2answers
93 views

Integrate a periodic absolute value function [duplicate]

\begin{equation} \int_{0}^t \left|\cos(t)\right|dt = \sin\left(t-\pi\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor\right)+2\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor \end{equation} I ...
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0answers
132 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 ...
0
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1answer
108 views

Period of $\sin(ax)+\cos(bx)$

Take the function $f(x)=\sin(ax)\cos(bx)$, with $a,b>0$. Suppose there are positive integer $p$ and $q$ such that $ap=bq=r$. Then $r$ is a period of $f$ but non always the shortest : for $f(x)= ...
2
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0answers
111 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 ?
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0answers
29 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 ...