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Questions tagged [periodic-functions]

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

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Principal value definition of the fractional Laplacian on the torus

I have a periodic function $u$ defined on the torus $\mathbb{T}^2= \mathbb{R}^2/\mathbb{Z^2}$ and want to find the corresponding definition of the fractional Laplacian $(-\Delta^{s})u$ in terms of the ...
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37 views

Periodic or non periodic function potting

I am currently working on plotting a function and figuring out if its periodic or not. The function is as follows: $$x_n = \beta\cdot x_{n-1}+\alpha \gamma\cdot \mathrm{sgn}(x_{n-3})+\alpha(1-\gamma)\...
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The limit of a function from different directions: rotate the function and take limit

Let $\{(x,y)\}_{x\in\mathbb R}$ be a continuous curve; what will be implied if it is given that the any rotated version of that curve have a limit? i.e. let $(s,t)=A(x,y)$ be a rotation around the ...
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Can the analysis of periodic functions be applied to the twin prime conjecture?

In my previous post, using the fact that $$(6n\pm 1)\in \mathbb P \iff n\ne 6ab\pm a \pm b$$ I proposed a sieve to identify suitable candidates for $n$, and by extension twin primes of the form $6n\pm ...
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2answers
44 views

Proof that $x\sin(x)$ has infinite accumulation points

I have to find a sequence with infinite many accumulation points and intuitively I thought about $\sin(x)$ - since it is periodic, it has points from its codomain that get repeated infinite many times....
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1answer
23 views

How to find out the zeros of the function?

Let $f : \mathbb R \to \mathbb R$ be a continuous $2\pi$-periodic function, i.e. for every $t \in \mathbb R$, we have $f(t) = f(t+2\pi)$. Prove that there exists $t_0\in \mathbb R$ such that $$f(t_0) =...
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2answers
29 views

Trying to solve simple pde $u_t = iu_{xx}+2iu$

I'm trying to solve $u_t = iu_{xx}+2iu$ where we know $u(0,x) = \cos(2\pi x)-i\sin(2\pi x)$, $0 \leq x < 1$, $0 \leq t$ with periodic boundary conditions. This is what I tried: Assume $u(t,x) = T(...
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31 views

Grade 12 Functions

I was hoping someone can explain to me step by step to get the answer for the following equation, I have no clue how to even begin solving the following questions. What is the period of this ...
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1answer
34 views

Fourier series of $f(x)=\int\limits_0^x \ln\sqrt{\frac{1}{2}\left| 1+\sqrt3 \tan\frac{t}{2} \right|} \ \text dt$

Find the Fourier series of the function $$f(x)=\int\limits_0^x \ln\sqrt{\frac{1}{2}\left| 1+\sqrt3 \tan\frac{t}{2} \right|} \ \text dt$$ or show that it does not exist. The first thing I have ...
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33 views

Show that $f$ is periodic, where $\int_{x}^{x+a}f(t)dt=b$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ continuous, for which there is an $a>0$, $b\in\mathbb{R}$ such that $\int_{x}^{x+a}f(t)dt=b$, any $x\in\mathbb{R}$. Show that $f$ is periodic. By $F(x)=\int_{...
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1answer
16 views

Fundamental frequency of the sum of two sinusoids

I'm trying to find the fundamental frequency of $\cos(23t + \pi/2) + \sin(5t + \pi/5)$ Now the period for both are $2\pi/23$ and $2\pi/5$ respectively. So i need to find the LCM of these two ...
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1answer
41 views

Why is the fundamental period of $\sin^3(2t)$ given by $1/\pi$ rather than $\pi$?

Take a look at the following function: $$ x(t) = \sin^3(2t) $$ In order to show the periodicity of the signal, we need to prove the following equality $$ x(t) = x(t+T) $$ We first use some ...
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Literature request - finite Fourier series expansion of bounded, periodic, differentiable functions?

Question - Is someone aware of a published work proving the following: Every bounded, differentiable, periodic function of $n$ variables has a Fourier Series expansion with exactly $m$ terms ($m$ is ...
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1answer
43 views

Can the sum of two periodic functions over $\mathbb R$ be strictly increasing?

If there exists a direct sum decomposition for the real space $\mathbb R=V\oplus W$, and define $f(v+w)=v$, $g(v+w)=w$, then $h(v+w)=v+w$ is strictly increasing. But I can't find explicitly one such ...
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32 views

Showing that a periodic solution exists using a sinusoidal logistic equation.

My equation of interest is a logistic growth model affected by a sinusoidal oscillation. $$ \frac{dN}{dt}=(rN(c-N)-\mu N)(1+cos(2\pi t)) $$ I want to show a periodic solution exists so I was thinking ...
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1answer
27 views

Counterexamples to equivalences between $L^n$-convergence types on periodic function spaces.

I am working with $\mathbb{Z}$-periodic continuous functions from real to complex numbers, and I am comparing $L^2$, uniform and pointwise convergence. We have defined $$\|f_n - f\|_{L^2} = \sqrt[2]{\...
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26 views

If f'(x) is periodic then f(x) is also periodic? [duplicate]

I saw the proof that if $f(x)$ is periodic, then $f'(x)$ is also periodic But then I wondered if the reciprocal was true, if $f'(x)$ is periodic then $f(x)$ is also periodic? I don't know if I can ...
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1answer
31 views

Why is this function periodic for only certain values of k.

Lets start by defining the function in question; $y(t)= \frac{\pi{sin(kt)}-ksin{(\pi{t}})}{\pi^{2}k-k^{3}}$ Now the question. So from what i gather the function is periodic when $k=n\pi\;$ , $\;\...
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1answer
77 views

Fourier series of non-periodic function $f(x)=e^{-\frac{ax}{L}}$

The definition of Fourier series states that It decomposes any periodic function or periodic signal into the weighted sum of a (possibly infinite) set of simple oscillating functions, namely sines ...
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2answers
27 views

Cauchy-Schwarz inequality for $L^2$-norm on periodic functions space

I have proven something that is definitely not true (Lemma 2), which is why I am intersted where I err. Definition Let $C(\mathbb{R}/\mathbb{Z},\mathbb{C})$ be the set of all continuous $\mathbb{Z}$-...
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Finding the Period/Frequency from data without plotting it.

I'm in basic trigonometry and currently learning how to find equations having been given the data only. I understand the concept pretty well, but I usually make a program on a TI-84 for solving my ...
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55 views

Hyperelliptic function addition formula

$$x= \int_{0}^{f(x)} \frac{du}{\sqrt{1-u^2}\sqrt{1-k^2u^2}\sqrt{1-l^2u^2}}$$ $$f(0)=0$$ If we apply derivative operation for both sides, we get: $$f'(x)=\sqrt{(1-f^2(x))(1-k^2f^2(x))(1-l^2f^2(x))}$$...
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1answer
38 views

If $r(t)$ is periodic and non-constant, then there exists a minimal $T$, such that : $r(t+T) = r(t)$. [duplicate]

Exercise : Let $r(t)$ be a periodic and non constant function. Prove that there exists a minimum $T \in \mathbb R$ with $T>0$ such that $r(t)$ is $T$-periodic, meaning that $T$ is the ...
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3answers
33 views

Periodic function such that integral of $|x(t)|$ is finite

I'm wondering if there is a periodic function such that the integral of $|x(t)|$ is finite. I tried with various functions, but the result is always infinite. Thank you in advance.
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1answer
58 views

Continuous periodic function with Fourier series behaving like $1/n$?

Is it possible to have a continuous periodic function whose Fourier series $(a_n)_{n\in \mathbb{Z}}$ is such that $$ n . a_n \rightarrow 1?$$ I know that there exist continuous periodic functions ...
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3answers
50 views

how to Find the period of complex exponential function?

How we can find period of this sequence? $x[n]= e^{jn2π/3}$ Is it equal to $T=2π/(2π/3) $ or not? I mean relation $T=2π/\omega$ will be valid in this case or not?
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2answers
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Showing that integral is related to sine function in elementary means

So I am trying to prove the reflection formula for the gamma function by showing that $$\int_{0}^{\infty} \frac{v^{s-1}}{1+v}dv=\frac{\pi}{\sin(\pi s)}$$ for $0 < \Re(s) < 1$ , as these two ...
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1answer
22 views

Infimum of positive periodic function is positive

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a positive, smooth, periodic function with period $T$. I need to show that the infimum of $f$ is positive. My idea is that on each interval of a period $f$...
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26 views

Solving a system of differential equations and finding periodic solutions

Consider the following system of differential equations: $\frac{dx}{dt}=y-1$ $\frac{dy}{dt}=-xy$ So I figured that, if the initial conditions satisfy $(x_0,y_0)\ne(0,1)$, I can rewrite the ...
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A clarification of Fourier representation of a function

As I was studying Fourier representation of a $T$-periodic function $f(t)$, I came up with two different definitions. For a function with discrete domain, $f(t) = \sum\limits_{k=0}^{T-1} C_k e^{i2\pi ...
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1answer
33 views

Periodic solution of PDE

I am little confused in how to solve this equation : $$u +u^{''}=0$$ where $u$ is real $2\pi-$periodic function and the derivative is in the sens of distributions So my question is that what is the ...
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2answers
86 views

What is the period of $x\sin x$?

What is the period of $x\sin x$? I’m not able to solve after $$(x+t)\sin(x+t)=x\sin x$$
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44 views

Fourier representation of a function with changing period

I learned that the Fourier representation of a periodic function $f(t)$ with period $T$ is $\sum \limits_{k=-\infty}^{\infty} C_k e^{i2\pi k t/T}$ with appropriate constants $\{C_k\}$. My question: ...
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References for a proof of a Jackson's inequality?

Let $g:[0,2\pi]\to \mathbb{C}$ which is $\mathcal{C}^k([0,2\pi],\mathbb{R})$ and periodic. If $\mid f^{(k)}(x) \mid\le 1$ then for each $n\in \mathbb{N}^*$, there exists a trigonometric polynomial $T_{...
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2answers
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Fourier Analysis - Functions on a Circle

In general, if a continuous function $g(x)$ is defined on the interval $[-\pi,\pi]$, can I say it is definitely possible to extend this function to be a $2\pi$-period function? I think we need to make ...
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33 views

Can we apply IVT to solve this?

Let $f:[0,2\pi] \to \mathbb R$ be continuous and periodic (meaning that $f(2\pi) = f(0)$). Show that there exists an $x ∈ [0, π]$ such that $f(x) = f(x+\pi)$. Any hints?
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124 views

'Spiky Periodic Things' - Do these objects have a name, and is there a method for finding the boundary curves?

This question was originally about evaluating the sum $\sum_{n=0}^\infty e^{nix}$, but I figured out the answer about half way through writing it. So instead, I decided to ask a slightly different ...
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1answer
35 views

Determine the period for the function y=10cot(10π/11x−7π/11) [closed]

Determine the period for the function $y=10\cot(\frac{10π}{11x}−\frac{7π}{11})$
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37 views

Proof periodic equation

I don't understand how he get $b- \sqrt2, -a, -b$.... and what is the periodic function?
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106 views

If $f(x+1)=f(x)$ then?

Let $f: \ \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x+1) = f(x)$, $\forall x \in \mathbb{R}$. Then which of the following statement(s) is/are true? $f$ is bounded. $f$ is bounded ...
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1answer
18 views

What is the period of this signal?

Below is the signal : $y[n] = j ^ n$ Someone told me that the period is 4 ,but he didn't explain me why. Can anyone help me ?
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1answer
36 views

Integrating a nested periodic function

Suppose $g(x)$ is a differentiable, real-valued, periodic function with period $a$ such that for all $u$, $\int_u^{u+a} g(x)dx=0$. Is it true, then, that $\int_u^{u+a}g(x+g(x))dx=0$ for all $u$?
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68 views

Show that $\lim_{n\to\infty} \int_{0}^{1}f(x)g(nx)dx = 0$ $g$ periodic on $\mathbb{R}$

Let $g$ be a continuous periodic function on $\mathbb{R}$ with $g(x + 1) = g(x)$. Assume that $\int_{0}^{1}g(x)dx = 0$. (a) Let $f$ be continuous on all $\mathbb{R}$ Show that $\lim_{n\to\infty} \...
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2answers
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How can I prove that this function is periodic :

I know that a periodic function satisfies $F(x+a)=F(x)$ $y=\left\lfloor\frac{\sin x}x\right\rfloor$ And the period of this fuction is $2\pi$ But putting $f(x+2\pi)$ I can't evaluate denominator ...
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Find the periodicity with the help of Laplace transform

I have a function $$x(t) = \pi\cos(21\omega_0t)+0.1\cos(39\omega_0t)$$ that I want to solve T from the periodicity identity, $x(t)=x(t+T)$. What I have tried now is basically just solving $x(t)=x(...
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1answer
46 views

trapezoid approximation of sine function

I want to find a reasonable good periodic trapezoid function approximation for the sine (or cosine) function. My use case is it to divide the year in 4 epochs: visible light length stays roughly ...
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25 views

Given a function which is both periodic and continuous, show there exists the smallest period. [duplicate]

We are given a function $f$ which is periodic and continuous. We also know that $f$ is not a constant function. We need to show that there exists a period that is the smallest period. I tried this ...
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1answer
91 views

Find all functions satisfying $f(x+1)=\frac{f(x)-5}{f(x)-3}$

Find all functions satisfying $$f(x+1)=\frac{f(x)-5}{f(x)-3}$$ My try: We have $$f(x+1)=1-\frac{2}{f(x)-3}$$ Letting $g(x) =f(x+1)-3$ We get $$g(x+1)=-2-\frac{2}{g(x)}$$ Any clue here?
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45 views

How To solve this type of inequation

Let $f$ be a regular $2\pi-$periodic function . How to find the functions that solve this inequality $a \le f+f^{'}+f^{"} \le b$ where $a,b \in \mathbb{R}_{+}$ . Could you give me some books to ...
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1answer
43 views

Calculating a periodic signal (way of solving this)?

I created my own examples so i can have the gist of how to solve the real ones that my homework needs so here we go: $$x(t)=\sum_{n=-\infty}^\infty \Pi\left({t-4n\over2}\right) + \sum_{n=-\infty}^\...