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

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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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34 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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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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63 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 UPDATE: The question has been superseded by Single-statement Continuous Periodic function without trigonometry ...
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34 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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38 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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31 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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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
47 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 ...
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45 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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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 ...
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30 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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39 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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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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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 ...
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1answer
55 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? ...
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72 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 ...
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1answer
77 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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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!
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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, ...
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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 ...
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71 views

What does $\Bbb R/2\pi$ for a set mean?

I simply cannot figure out what this means. I read this on an article about the scalar product of $2\pi$ periodic functions. it says that < f,g > goes from $\Bbb R/2\pi \to \Bbb C$ (complex) Do ...
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4answers
182 views

Proving a function is continuous and periodic

Suppose we are given a function $$g\left ( x \right )= \sum_{n=1}^{\infty}\frac{\sin \left ( nx \right )}{10^{n}\sin \left ( x \right )},x\neq k\pi , k\in\mathbb{Z}$$ and $$g\left ( k\pi \right ...
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Why is $\int\limits_{u}^{u+\omega_j} f'(z)/f(z) dz \in 2 \pi i \mathbb{Z}$?

Because $f(u) = f(u+\omega_j)$ for $j \in \{1,2\}$ it applies $$\int\limits_{u}^{u+\omega_j}\frac{f'(z)}{f(z)}dz \quad \in 2\pi i \mathbb{Z} \quad \text{for} \quad j=1,2, $$ Hello, I write my thesis ...
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Scalar product with $f(x,y)$ when $ \dot{X} = f(X)$ has periodic orbit

Let $g(x,y), f(x,y) \in C^1: R^2 \to R^2$ such that $f(x,y) \cdot g(x,y)=0$, $\forall (x,y).$ Prove that if $\dot X = f(X)$ has a periodic orbit, then $g$ have a root Intuitively I can see that the ...
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61 views

Integral of Absolute Value of $\sin(x)$

For the Integral: $\int |\sin (ax)|$, it is fairly simple to take the Laplace transform of the absolute value of sine, treating it as a periodic function. $$\mathcal L(|\sin (ax)|) = ...
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about Poincare map

I saw that the Poincare map is defined by the flow of the periodic system with the least period $T$. that is, $$P(x):=\phi_T(x)$$ is a Poincare map with flow $\phi$ of time $T$. but I think if we ...
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Odd periodic Function

Given an odd function $f$, defined everywhere, periodic with period $2$, and integrable on every interval. Let $g(x) = \displaystyle\int_{0}^{x}f(t)dt$. I know that ...
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1answer
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Find period of the Function

The given question $$f(x) = \sqrt{\frac{8}{1+x} + \frac{8}{{1-x}}}$$ $$g(x) = \frac{4}{f(\sin x)}+\frac{4}{f(\cos x)}$$ find period of $g(x)$? What I have done putting $\sin(x)$ and $\cos(x)$ in ...
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Help with Proving : Period estimation for for concatenated sequences

Assuming I have two 8-bit random number sequences $s[n]$ and $d[n]$ which each have a period of $X$ and $Y$ respectively. Therefore: $$s[n+X] = s[n]\\ d[n+Y] = d[n]$$ If they were concatenated ...
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Defining periodic functions?

Consider $f: (a,a+p] \rightarrow \mathbb{R}$. What is the "formula" for the p-periodic function $g$ which has the property that $g(x + np) = f(x)$ for all $x$ in $(a, a+p]$? I am well aware of how ...
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36 views

Function with Multiple Periods

Basically I'm trying to fit some data with seasonal effects to a periodic function, and the problem I'm running into is that the local minima usually occur around April, and the local maxima usually ...
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Solving differential equation with Fourier-series-inhomogenity

Let $\lambda$ be a real number , $(c_k)$ a complex sequence with $\mid c_k \mid \leq C(1+\mid k \mid)^{-2}$ for all k with a constant $C \geq 0 $. Find all periodic, two times differentiable ...
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Proving a real valued function is periodic, and sketching it using obtained information

Consider an arbitrary function, something like $f\left ( x \right )=\arccos \left ( \sin \left ( 4x \right ) \right )$. Its graph looks like this: I was greatly confused by the image below, because ...
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2answers
87 views

All solutions to functional equation $f(x+1)-f(x)=1$

I was thinking of the possibility of finding all solutions other than $f(x)=x$ for the functional equation: $f(x+1)-f(x)=1$ If there are other solutions, what will be some restrictions for the ...
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Find a function $g(x)$ satisfying the above conditions.

Find a function $g(x)$ satisfying the above conditions:- a)domain is $(-∞,∞)$. b)range is $[-2,8]$. c)$g(x)$ has a period $π$. d)$g(2)$=3. ATTEMPT: Since the function is periodic with period $π$ ...
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Finding period of a function? Am I doing something wrong?

I need to find the period of the function $$\large{\frac{\sin (\sin {nx})}{\tan{\frac{x}{n}}}}$$. According to me, the period of $\sin (\sin (nx))$ should be $\large{\frac{2\pi}{n}}$ and the period ...
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Periodicity of trigonometric functions directly from their power series

My question is very simple yet I've gotten nowhere with it. Is there any way one can, without directly or indirectly referencing any differential equations satisfied by the circular trigonometric ...
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71 views

“Perimeter” of the sine function

Given a sine function with certain parameters (period, amplitude) I would like a function to calculate its "perimeter", i.e. the length of the curve itself. Everyday application: let's say we need to ...
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1answer
42 views

How to simplify this summation, or express as integral? $\sum_{t=-\infty}^\infty\frac1{\sqrt{(t+ax)^2+4}}$

How to simplify this summation, or express as integral? $$f(x)=\sum_{t=-\infty}^\infty\frac1{\sqrt{(t+ax)^2+4}}$$ $a$ is a constant, say, 24. $$f(x)=\sum_{t=-\infty}^\infty\frac1{\sqrt{(t+24x)^2+4}}$$ ...
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44 views

Periodic function

Considering the function $F(x)=x-E(x)$ such as $E(x)$ is the integer part of $x$ So here just with observation we can see that : $f(x+1)=(x+1)-E(x+1)=f(x)$. But here mathematics is not based just ...
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27 views

Find the rate of change of main dependent variable

We have $f:\mathbb{R}\rightarrow\mathbb{R},\:f(x)=x^2+x\sin(x)$, and we need to find intervals of monotonicity. Here is all my steps: $f'(x)=2x+x\cos(x)+\sin(x)$ $f'(x)=0 \Rightarrow x=0$ the only ...
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267 views

Can a multiply-periodic complex function be analytic?

It's possible to construct complex periodic functions with two periods in different directions, such as $f(z) = \cos x + i \sin 2y$. That has periods $2\pi$ and $\pi i$. It's also not analytic. ...
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32 views

Maximum period-length in decimal chain-addition?

Decimal chain-addition $^*$ takes a finite initial "seed" string of decimal digits, say $x_1x_2...x_n$, and defines an infinite periodic string $x_1x_2...x_n...$ by iterating the following rewrite ...
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63 views

Algebraic Curves and Second Order Differential Equations

I am curious if there are any examples of functions that are solutions to second order differential equations, that also parametrize an algebraic curve. I am aware that the Weierstrass $\wp$ - ...
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76 views

Finding this weird limit involving periodic functions with periods 5 and 10.

If $f(x)$ and $g(x)$ are two periodic functions with periods 5 and 10 respectively, such that: $$\lim_{x\to0}\frac{f(x)}x=\lim_{x\to0}\frac{g(x)}x=k;\quad k>0$$ then for $n\in\mathbb N$, the value ...
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1answer
58 views

Integral of periodic function.

I was wondering if anyone can help me with this, if f(x) is a periodic function with period T then it satisfies $$\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)dx$$ for all $a \in \Bbb R$. It is clear that ...
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37 views

Is this function $2\pi$-periodic?

If I construct $$G(x) =\sum_{1}^{\infty} g(x +2n\pi),$$ does this make $G(x)$ $2\pi$-periodic? My understanding is that if $G(x)$ were now $2\pi$-periodic, then that means $G(x) = G(x + 2\pi$) = G(x ...
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1answer
33 views

How to calculate power of a non-continuous signal

I have to find the power of the following signal and would like to know if I'm doing this right or, if I'm doing it wrong, how to do it. The equation for power in my textbook is $\overline{m^2(t)} = ...