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

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Prove or disprove: If $ f $ is periodic, then $|f|$ is also periodic.

I started with the definition of periodic function and absolute value function. And I do it with discussing different cases of $x$ and $p$. But I got stuck with when $ -p\leq x <0 $ , I want to ...
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18 views

References about functions of the following form $f(t)=f(t+P(t))$

Where I can find any references about real functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(t)=f(t+P(t)),\forall t\in \mathbb{R}$ ?
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5 views

Why are the points of a period module necessarily isolated?

Given a non-constant meromorphic function $f(z)$ then how can we say that the absence of a finite accumulation point of our module $M$ implies that $f$ is constant? I understand that an accumulation ...
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40 views

Integrals of a nonnegative, symmetric, periodic, continuous function

Let $f:\mathbb{R}^2\to\mathbb{R}$ be such that $f$ is nonnegative, symmetric, $\ell$-periodic in both variables, zero exactly on the diagonal of its domain, and continuous. Specifically, $f$ has the ...
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0answers
46 views

When talking about periods of a function why is it common to introduce a factor of $2$?

In many books I have read, when talking about periodic functions, they tend to write the period as $2\omega$. Why do we need the $2$? I haven't come across any working where the factor of $2$ is ...
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1answer
74 views

proving periodic function f is uniformly continuous [duplicate]

So I'm new to uniform continuity and this is just an exercise that was scribbled on the board in my real analysis class. But it's is a tricky question for me: if $f:R\to R$ is periodic with period P ...
3
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2answers
89 views

Prove that $|f(x)-f(y)|\le|x-y|$ for all $x$ and $y$.

Let $f(x) = |x|$ for $x \in [-1,1]$ and extend $f$ to all of $\mathbb R$ by requiring $f(x+2)=f(x)$. Prove that $|f(x)-f(y)|\le|x-y|$ for all $x$ and $y$. This is easy to observe on a graph but my ...
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1answer
47 views

How can $e^x$ be restated for small $x$?

Suppose I have the following equation: \begin{equation} 1=\frac{S_0}{\gamma}(1-e^{-\gamma T_n})+\lambda\int^{t_{n+1}}_{t_n}e^{-\gamma(t_{n+1}-\tau)}g(\tau)d\tau. \end{equation} If I make two ...
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0answers
36 views

Periodic boundary conditions: image summation vs periodic solution

Let's say we have a linear PDE and its solution is the field $\sigma(x,y)$. I want to use this field in a simulation, and in order to avoid boundary effects, I want to use Periodic Boundary ...
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1answer
10 views

Is there a natural, circular (toric) Gaussian density?

I am looking for density distributions over a circular set (think about $[-\pi, \pi[$ or $\Delta+[0, T[$ in general $\forall T \in \mathbb{R}^{+*}, \Delta \in \mathbb{R}$). Is there a density ...
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3answers
83 views

Closed periodic curve

Let $\textbf{$\gamma$}: \mathbb{R} \rightarrow \mathbb{R}^n$ be a smooth curve and let $T \in \mathbb{R}$. We say that $\textbf{$\gamma$}$ is $T$-periodic if ...
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3answers
43 views

A function that is zero for every integer multiple of $k$

I'm new to this kind of question so it may be a trivial one but I can't find a general solution: I'm searching for a function that has a value of $0$ for every integer multiple of some fixed real ...
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1answer
36 views

Proof: sum, multiplication and division of two periodic function

Proof that the sum, multiplication and division of two periodic function with the same period, is again a periodic function Please help, I need these three proofs for my calculus homework
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1answer
19 views

Fundamental matrix of $y''+\epsilon f(t)y=0$

I converted this ode into a linear matrix form like $y'=Ay$ and tried to solve this, but I couldn't find a fundamental solution which satisfies $\Phi (0)=I$, which is required in one of my assignment ...
3
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2answers
105 views

secret formula for the “sin” wave with variable rising/falling edge

My math is pretty much forgotten. I was wondering if someone can take a look at this and share what's the formula for creating something like this. ...
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1answer
35 views

Passage in a proof from Hofer-Zehnder

The proof I'm referring to is to the following theorem. Assume $S$ is a compact regular and strictly convex energy surface for the Hamiltonian field $X_H$ in $\mathbb{R}^{2n}$. Then $S$ carries a ...
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1answer
64 views

How to calculate the period of a nonlinear Diff Eq?

I have this Diff Eq: $$2g''+g'^{2}+ag+a-b=0 $$ We can manipulate this into an equivalent relation: $$g'^{2}=ce^{-g}-ag+b $$ ...which makes c a conserved quantity: ...
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1answer
28 views

Why do the periods of a meromorphic fuction form a discrete module?

I've come across this statement: If $f$ is a nonconstant meromorphic function, the module $M$ containing all its periods cannot have an accumulation point, since otherwise $f$ would be a ...
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22 views

The number of periodic solutions of Abel differential equations

I'm asked to proof the number of T-periodic solutions of ODE $$x'=p_0(t)x^3+p_1(t)x^2+p_2(t)x+p_3(t)$$ is at most three, where $$p_i(t)$$ is continuous functions with period T (i=0,1,2,3) and ...
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2answers
35 views

Evaluating a definite integral of a Bessel-type function

I have an expression as follows: $\int_{0}^{2\pi} \sin{(x\sin{(\theta}) - n\theta)}\mathrm{d}\theta$ For real $x$ and $\theta$ and positive integer $n$. From plugging it into Mathematica with ...
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4answers
146 views

Show that $f(x):=\cos(x^2)$ is not periodic.

How can I proof that the following function $f(x):=\cos(x^2)$ is not periodic? I think that I should find the zero points of the function but I don't know how to calculate it. Thank you very much ...
5
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2answers
64 views

Let $f\colon\Bbb R\to \Bbb R-\{3\}$ be a function such that there exist $T>0$ with $f(x+T)=\frac{f(x)-5}{f(x)-3}$ for every $x\in\Bbb R$.

Let $f\colon \Bbb R\to \Bbb R-\{3\}$ be a function with the property that there exist $T>0$ such that $f(x+T)=\frac{f(x)-5}{f(x)-3}$ for every $x\in \Bbb R$. Prove that $f(x)$ is periodic and find ...
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1answer
54 views

What is the fundamental period of $f(x)=\frac{|\sin x|+|\cos x|}{|\sin x -\cos x|}$

What is the fundamental period of $f(x)=\frac{|\sin x|+|\cos x|}{|\sin x -\cos x|}$ I tried it. $f(x)=\frac{|\sin x|+|\cos x|}{|\sin x -\cos x|}=\frac{|\sin x|+|\cos x|}{\sqrt 2|\sin ...
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2answers
54 views

Solution to harmonic oscillator with periodic forcing

Using matlab symbolic processor, I can get the homogenous and particular solution to a harmonic oscillator with periodic forcing. I'm trying to write the particular solution in a compact form, with ...
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3answers
39 views

Proof that any integer of the fundamental period, is also a period of the function?

How can it be proved that for a given $f(x)$ with a period $p$, that for any integer $n=1,2,\dots$, the product $np$ is also an period of the function $f(x)$? I know that the definition of a periodic ...
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4answers
101 views

Period T of function $y = \cos(nx)$

There are two functions: 1) $f(x) = \cos(nx)$ 2) $f(x) = \cos(x)$ $T=2 \pi$ is the fundamental period of $(2)$ function. $T_1$ is the fundamental period of $(1)$ function. How to prove that ...
0
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1answer
31 views

Fourier Sine Series -The region where the function is defined and its period

I am pretty new to calculus, and I am trying to understand some basic rules to solve Fourier Series. I don't know how I deal with the region where a function is defined and its period. For instance: ...
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2answers
29 views

Does the arccos itself contain all solutions or just one solution?

For the equation $$\cos(x)=\frac{1}{2}$$ All solutions are: $$x=\pm\frac{\pi}{3}+p2\pi,\quad p\in\mathbb Z\:.$$ To find these solutions, I use the inverse cosine ($\arccos$ or $\cos^{-1}$). Is the ...
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0answers
49 views

Does Every Periodic Function Have An Associated Differential Equation?

My question is the opposite to proving the existence of periodic solutions to ODE's. Assume that $\ f(z)$ is a periodic function over the $\mathbb{R}$ , or doubly periodic over some lattice $\Lambda$ ...
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20 views

With 2 as smallest period of the function $f(x)$= $\tan^2[(\frac{\pi x}{n^2-5n+8})]$ + $\cot(n+m)\pi x$ ;the period m can't belong to is?

Here n $ \in N$ , m $\in Q$. Options are: A) $(-\infty, -2) \cup (-1, \infty)$ B) $(-\infty, -3) \cup (-2, \infty)$ C) $(-2,-1) \cup (-3,-2)$ D) $(-3, -5/2) \cup (-5/2, -2)$ I have an answer to ...
5
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1answer
53 views

Determining if a sum of trig function is periodic

$$2 + \sin(2\pi\cdot t) + 3\cos(3\pi\cdot t) - 5\sin(7t-\frac{\pi}{4})$$ Is there any manual, easy, way of knowing such a function is not periodic? I'd love to know if there's any method which ...
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27 views

Single-statement Continuous Periodic function without trigonometry and complex numbers

Superseding the question Periodic function without trigonometry and complex numbers , I am now asking: Can I get a single-statement continuous periodic function without using trigonometric functions ...
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2answers
78 views

Periodic function without trigonometry and complex numbers [closed]

Can I get a periodic function without using trigonometric functions or complex numbers? UPDATE: The question has been superseded by Single-statement Continuous Periodic function without trigonometry ...
3
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1answer
79 views

Double integral definition of periodic Sobolev spaces

Preliminaries: For $s \geq 0$ define the periodic Sobolev spaces $H^s(\mathbb T)$ with norm $$ \| f \|_{H^s(\mathbb T)}^2 = \sum_{k \in \mathbb Z} \big(1 + |k|^2 \big)^s \, \big|\widehat{f}_k \big|^2, ...
0
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1answer
47 views

How to derive the complex Fourier series of $s(t) = 1-e^{-2t}$? [closed]

I have the periodic function $s(t)=1-e^{-2t}$. I am required to derive the complex Fourier series of $s(t)$. I have some knowledge of Fourier series but not enough to know if I am doing it correctly. ...
2
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1answer
131 views

Is there a proven formula for Pisano periods (Fibonacci numbers taken modulo $n$)?

I've seen on wikipedia this formula: $$\pi_k(p^n) = p^{n-1}\pi_k(p) $$ It says that the formula holds unless p is k-Wall-Sun-Sun prime, or k-Fibonacci-Wieferich prime, that is, $p^2$ divides ...
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19 views

How does emergent oscillation appear in this animation of concentric circles?

I made this animation and I barely understand it. http://bl.ocks.org/tophtucker/500d2a010105cfcc87db It's a bunch of concentric circles with exclusion compositing. The radius of the ...
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1answer
61 views

Evaluating the integral $\int_{-\infty}^{\infty}{e^{2i\mu t}\frac{{\sin^2{\lambda t}}}{\lambda t^2}}dt$

It is stated that (for $\lambda>0$) $$\frac{1}{\pi}\int_{-\infty}^{\infty}{e^{2i\mu t}\frac{{\sin^2{\lambda t}}}{\lambda t^2}}dt = 1-\frac{|\mu|}{\lambda}$$ for $ 0\leq|\mu|\leq\lambda$, and zero ...
3
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1answer
70 views

Integration: The Periodic Function and its Properties

I am aware that if $f(x)$ is a periodic function with period $T$ then: 1.) $\int^{nT}_{0}f(x)\,dx = n\int^{T}_{0}f(x)\,dx$, for $n$ an integer. 2.) $\int^{T+a}_{a}f(x)\,dx = \int^{T}_{0}f(x)\,dx$, ...
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52 views

How to find a stable periodic solution of a nonlinear non-autonomous second-order dynamical system?

The system I am working on is in the following form: $$ \dot{x}=f(t,x,a)=f(t+T,x,a), $$ where $f(t,x,a) \in R^2$ is nonlinear and periodic in $t$, $T$ is a known constant, and $a$ is a vector of ...
0
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1answer
33 views

$f$ is smooth and periodic function,$\exists \lambda$ such that $f^{(4)}=\lambda f$ prove:$\exists \lambda$ such that $\lambda= (\frac{2\pi n}{T})^4$

Given $f:\mathbb{R} \rightarrow \mathbb{C}$ is smooth and periodic with period $T>0$ and exists $\lambda \in \mathbb{C}$ such that $f^{(4)}(x)=\lambda f(x)$ for any $x \in \mathbb{R}$. prove: ...
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2answers
88 views

Show that an irrationally periodic function is also a constant function [duplicate]

Let $f:\mathbb R \to \mathbb R$ be a function such that for any irrational number $r$, and any real number $x$ we have $f(x)=f(x+r)$. Show that $f$ is a constant function.
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32 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}= ...
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0answers
28 views

Examples of periodic functions without sin or cos in their formulations? [duplicate]

I'm trying to find a function f(x) such that NCspline (in Matlab) provides a better interpolation than does the Lagrange polynomial. But this function cannot be a linear combination of the sine and ...
4
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1answer
125 views

Finding period of a recursive sequence defined by modular operator?

If $f(x)$ is defined as : $f(x)=i$ ,if $x\equiv i$ (mod n), $0\leq i< n$ How can I prove whether the following recursive sequences are periodic or not? ...
3
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1answer
78 views

Period of Fibonacci mod $b$?

It is not too difficult to show that the Fibonacci numbers mod $b$ form a periodic sequence. I would like to say something interesting about the period. There is a small shortcut to the brute-force ...
1
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1answer
81 views

Need to prove continuous periodic function of $\varphi (x) \equiv \psi(x)$

Question: Let two $\varphi(x) $ and $\psi(x)$ periodic and continous functions such that $$ \lim_{ x\to\infty}(\varphi(x)-\psi(x))=0, \quad x\in \mathbb{R}. $$ Prove that $$ ...
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2answers
146 views

Finding Period From Functional Equation

How to find the period of real valued function satisfying $f(x)+f(x+4)=f(x+2)+f(x+6)$ ? Note:Use of recurrence relations not allowed.Use of elementary algebraic manipulations is better!
5
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2answers
192 views

The sum of two continuous periodic functions is periodic if and only if the ratio of their periods is rational?

There are several questions here on MSE about the periodicity of the sum of two continuous, periodic functions. Here's what I know so far: It's obvious that if the ratio of the periods is rational, ...
6
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5answers
591 views

When a periodic function is squared (or cubed, and so on…) does it always lose its periodicity?

For instance $$\sin^{2}\left(-\frac{\pi}{6}\right) = \sin^{2}\left(\frac{\pi}{6}\right)$$ i.e., $\sin^2 (x)$ is an even function and loses the $2\pi$-periodicity of $\sin x $. Is this true in ...