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

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On Laplace transform of periodic functions

I recently bumped into this theorem regarding the Laplace transform of periodic function: Given a periodic function $f(t)=f(t+p)$, where $p$ is its period, then its Laplace transform is given by: $$\...
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0answers
34 views

Suppose that a sequence {$a_n$} is eventually periodic. Prove that {$b_n$} is eventually periodic, given that they are in some relation.

To clarify, a sequence {$x_n$} is called eventually periodic if there exist positive integer $r$ and $p$ for which $x_{n+p}=x_n$ for all $n\geq r$. The relation they have is that, (i). $b_{n+1}=b_n+...
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2answers
53 views

Property of function $\varphi(x)=|x|$ on $\mathbb{R}$

Define $\varphi(x)=|x|$ on $[-1,1]$ and extend the definition of $\varphi(x)$ to all real $x$ by requiring that $\varphi(x+2)=\varphi(x).$ How do you prove that for any $s,t$ $$ |\varphi(s)-\...
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0answers
39 views

Does linear combination of periodic functions is periodic?

Does linear combination of periodic functions is periodic? I know that if $f$ is periodic and $g$ is periodic then $f+g$ is periodic. But what about the linear combination of two periodic functions??...
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1answer
97 views

If $g(x):=f(x^2)$ is uniformly continuous, is $f$ a constant (assuming it's periodic)?

I know that $f:\mathbb{R}\to\mathbb{R}$ is continuous and periodic. How can I prove that $f$ is a constant whenever $g(x):=f(x^2)$ is uniformly continuous?
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0answers
28 views

Are there entire functions that are unexpectedly periodic? [closed]

I wonder if there have ever been entire functions discovered/defined that turned out to be periodic and the periodicity was unexpected and surprising to mathematicians ?
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1answer
34 views

How to find the period of this trigonometric function

$y$ = $|\sin x|$ I know the period is π by drawing the graph, but I can't prove it. Please use this method we have learnt for other functions. For example $y=\sin2x$ $\sin2x= \sin2(x+T)$ $2x+2π=2x+...
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1answer
40 views

Integrating over discontinuities

I have the following integral: \begin{align} \int^T_0e^\tau I(\tau+t^0)d\tau. \end{align} In this integral, $I(t)$ is a function with period $T$. At each time $T, 2T, \ldots$, $I$ is increased by a ...
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1answer
40 views

If the function $g: R\rightarrow R$ is periodic, then $a_n=g(n)$ is periodic.

If the function $g: R\rightarrow R$ is periodic, then $a_n=g(n)$ is periodic. Is the converse true? Give a proof or counterexample.
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0answers
23 views

Understanding the relationship between phase and frequency in a particular equation

Context I have the following equation: \begin{align} a&=1-e^{-\omega t/x}+\frac{m}{\sqrt{1+x^2}}\Bigg[\sin(\alpha+\omega t + \varphi)-\sin(\alpha+\varphi)e^{-\omega t/x}\Bigg]. \end{align} This ...
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1answer
41 views

Formula for periodic sequence with increasing period

The first values of the sequence are: $\lt1, 0, 1, 1, 0, 0, -1, -1, -1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, -...
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3answers
151 views

What is the period of $f(x) = \sin^4(x)+ \cos^4(x)$?

This is an elementary problem but I'm just not getting the right answer. My reasoning is as follows: The period of $g(x) = \sin^4(x)$ is $\pi$ and that of $h(x) = \cos^4(x)$ is $π$ as well, so the ...
2
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1answer
72 views

Are all solutions of $ f_n (x)^{2n} + f_n ' (x)^{2n} = 1$ periodic?

Let $n$ be a strict positive integer and $x$ is a complex number. Define $f_n(x)$ as one of the solutions to $$ f_n (x)^{2n} + f_n ' (x)^{2n} = 1$$ Where the derivative is with respect to $x$. Why ...
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1answer
68 views

Prove that a continuous function on $\mathbb{R}$ which is periodic attains its extreme values.

Prove that a continuous function on $\mathbb{R}$ which is periodic attains its extreme values. I don't know how to start this proof, but I know I have to use the extreme value theorem, continuity ...
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2answers
51 views

Are periodic functions stationary processes, e.g. y=sin(t)?

According to wikipedia, a stationary process (or strict(ly) stationary process or strong(ly) stationary process) is a stochastic process whose joint probability distribution does not change when ...
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1answer
23 views

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
48 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 ...
6
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1answer
82 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
44 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
70 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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1answer
46 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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0answers
49 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 ...
7
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2answers
553 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: ...
3
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2answers
144 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 ...
0
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1answer
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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0answers
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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0answers
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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0answers
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
81 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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0answers
39 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 Conditions....
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1answer
11 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 ...
5
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3answers
86 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 $$\textbf{$\gamma$}(t+T)=\textbf{$\gamma$}...
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3answers
46 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 $k$....
0
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1answer
41 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
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 ...
3
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2answers
110 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. https://drive.google.com/file/d/...
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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: $$c=\left(g'^{2}+ag-b\right)e^{...
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1answer
29 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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0answers
28 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 $$p_0(t)\...
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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
163 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
65 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 ...
1
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1answer
59 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 (x-\frac{\pi}{4})...
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2answers
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 ...
2
votes
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 ...