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

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Finding $\sin^3\frac{x}{2}\cdot\cos^7\frac{x}{3}$ periodicity

I need to find the period of $\sin^3\frac{x}{2}\cdot\cos^7\frac{x}{3}$ but couldn't find a nice method how? any ideas?
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1answer
45 views

Constant function and periodicity

If $f$ is constant function then every real number $p>0$ is its period. I was wondering is the converse true, that is: If every real number $p>0$ is period of a function $f : \mathbb R \to ...
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3answers
61 views

Function $f$ such that $f$ is non-periodic but $f(f(x))$ is?

Is there a "nice" example of a function $f$ such that $f(x)$ is non-periodic but the composition $f(f(x))$ is? By nice I mean that preferably it will be defined entirely on the domain $R$ and be ...
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1answer
78 views

Limit of $\int_E f(nx) dx$ for a $1$-periodic function $f$ on $[0,2\pi]$

Let $E$ be a measurable subset of $[0, 2\pi]$. Assume that $f \in C(\mathbb R)$ is $1$-periodic, i.e. $f(x + 1) = f(x)$. Compute $$\lim_{n\to\infty} \int_{E} f(nx) dx$$. Since $f$ is continuous on ...
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1answer
43 views

mathematical representation of a pure sinusoidal tone

Considering the case of a pure sinusoidal tone, e.g. the tuning A note at $440 \text{Hz}$, how can one mathematically represent the pressure wave resulting? For the sake of simplicity, I want to ...
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1answer
69 views

if |f| is periodic then f is periodic [duplicate]

Decide whether the following statement about a function f: R -> R is true. If |f| is periodic, then f is periodic. Give a proof or counterexample.
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45 views

How to make a function periodic?

I have a nice little equation here, $f\left(x\right)=\frac{4}{\pi ^2}\left(x+\frac{\pi }{2}\right)^2-1$, which ever so nicely approximates (with somewhat good accuracy), a period of the sine function, ...
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41 views

Littlewood-Paley Decompositions and Periodic Besov Spaces

I'm currently working on some problems in $2$-dimensional periodic space, and it seems that the framework of Besov spaces will be useful to me. Since we're working in periodic space, we can consider ...
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550 views

|f| is periodic implied f is periodic

I can think of an example where this wouldn't hold. Take 1,-1,1,-1,-1,1,-1,-1,-1,1,-1,-1,-1,-1. But I can also prove that the statement holds. Claim: $|f|$ is periodic then $f$ is periodic Proof: ...
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2answers
139 views

Is $x_k=\sin(k)$ eventually/periodic?

A sequence is eventually periodic if we can drop a finite number of terms from the beginning and make it periodic. $$x_k=\sin(k)$$ I think this is periodic since the function is periodic and it ...
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96 views

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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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
75 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 ...
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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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37 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
85 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
44 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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20 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 ...
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2answers
108 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
38 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
156 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 ...
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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
56 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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55 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
45 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
115 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 ...
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1answer
32 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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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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50 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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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 ...
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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
79 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 ...
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1answer
80 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, ...
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1answer
48 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. ...
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1answer
180 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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0answers
21 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
75 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 ...