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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27 views

If a function $f$ is $L$-periodic then $f'$ has $2$ zeros in $[0,L)$?

Let $f: \mathbb{R} \longrightarrow \mathbb{R} $ be a differentiable and odd function. If $f$ is periodic and the (minimal) period $L>0$, then $f'$ has $2$ zeros in $[0,L)$? For example, this ...
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2answers
31 views

Sine Parametric function exercise

Find the biggest negative value of $a$ , for which the maximum of $f(x) =sin(24x+\frac{πa}{100})$ is at $x_0=π$ The answer is $a=-150$, but I don't understand the solving way. I would appreciate if ...
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2answers
82 views

Prove that $0$ is the only $2\pi$-periodic solution of $\ddot{x}+3x+x^3=0$

Prove that $0$ is the only $2\pi$-periodic solution of $\ddot{x}+3x+x^3=0$. I don't know how to deal with this non-linear differential equation. I tried to consider $\ddot{x}(t+2\pi)+3x(t+2\pi)+x^3(t+...
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0answers
16 views

Is it always possible to get the direct function for a system of ODEs?

Is is possible for all ODEs to derive the direct formula? I'm wondering if there is some (very difficult) mathematical method to get the direct formula. As an example, consider the following system: ...
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1answer
26 views

Name for the center amplitude of a sine wave

I am trying to find the name for the value at the center of a sine wave $$ y(t) = A\sin (kx \pm \omega t+\varphi) + D $$ i.e. the offset $D$ of the wave in the $y$-direction. Wikipedia calls it Center ...
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1answer
73 views

If $c\in\mathbb R,$ then prove that $\sin(x)+\sin(cx)$ is periodic iff $c\in\mathbb Q.$

We know period of $\sin x$ is $2π.$ So period of $\sin cx$ will be $\frac{2π}{|c|}.$ Therefore period of $(\sin x+\sin cx)$ is: $\text{LCM of }\left(2π, \frac{2π}{|c|}\right)=\frac{\text{LCM of }(2π, ...
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27 views

How to find fundamental period?

Suppose we have a function $h(x) = f(x) + g(x)$ where $f$ and $g$ are periodic functions with fundamental period $T_1$ and $T_2$ respectively. We know that the least common integral multiple of $T_1$ ...
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1answer
37 views

Period of trig functions

How to prove the periodicity and then find the primitive period of summation, products and compositions of trig functions? Is it possible to prove that the primitive period of sine function is $2\pi$ ...
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2answers
39 views

Finding the function $f(x)$ through the given set of conditions

I got terribly stuck while solving this question recently Let $f(x)$ be a real valued function such that $$f(0)=\frac{1}{2}; \quad f(x+y) =f(x) f(a-y) +f(y) f(a-x).$$ I do not want the rigorous ...
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1answer
56 views

Existence of periodic solution of a differential equation

Let us consider $f=f(x,t):\mathbb{R^n}\times \mathbb{R}\to\mathbb{R^n}$ a $C^1$ function which is periodic in $t$. We know that, under the hypothesis $x\cdot f(x,t)<0$ (with $|x|>M$ and $t\in \...
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0answers
42 views

Heat Equation Periodic Boundary Conditions

I'm solving the heat equation on a ring of radius $R$. The ring is parameterised by $s$, the arc-length from the 3 o'clock position. Using separation of variables I have found the general solution to ...
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1answer
31 views

Determining the period of $\sin 2x +\sin\frac{x}{2}$ without using the LCM of periods

I tried to calculate period of function described as: $$y=\sin 2x +\sin\frac{x}{2}$$ but without using LCM of periods. From definition of periodic function we have: $$\begin{align} 0 &= \...
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1answer
28 views

Weierstrass Elliptic functions

Please suggest a good book on preliminary Weierstrss elliptic functions or some link from where I can learn about them in details. Please help.
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2answers
38 views

Find the fundamental period of $f(x)=\sin\left(2m\pi\{x\}\right)$ and $g(x)=\sin\left((2m+1)\pi\{x\}\right)$

Find the fundamental period of $f(x)=\sin\left(2m\pi\{x\}\right)$ and $g(x)=\sin\left((2m+1)\pi\{x\}\right)$, where $\{x\}$ denotes the fractional part of $x$ and $m$ is a natural number. Since the ...
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3answers
50 views

Periodic Functions, Why does T have to be greater than 0

If a periodic function can be describe has: $$\forall x\in\mathbb{R},\exists t>0, st. f(x +t) = f(x)$$ Why does 't' have to be greater than 0?
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1answer
44 views

Is $\displaystyle{\exp ^{z^2}}$ periodic?

Is $\displaystyle{\exp ^{z^2}}$ periodic? I was trying to find about the periodicity of any function of the form $\displaystyle{\exp ^{p(z)}}$, where $p$ is a complex polynomial. Since $\exp z$ is ...
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1answer
95 views

Possible periods of periodic sequences of reals obeying $x_{n+2} = 1+x_{n+1} x_n.$

If $\{x_n\}$ is a real sequence with period $T$ obeying $x_{n+2} = 1+x_{n+1} x_n,$ what are the possible values of $T$? $T=3$ is possible: $2, -1, -1, 2, \dots$. In fact, some analysis shows this to ...
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0answers
10 views

Wave equation: function 2L-periodic.

I have to prove that any function that is even with respect to x=0 and x=L is necessarily 2L-periodic. We are studying wave equations but I don´t know how to prove it. Can you give me a clue?
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48 views

Can Eulers product for $\sin x$ be generalised for any periodic function?

$$\sin x=x\prod _{n=1}^{\infty }\left(1-\left(\frac{x}{n\pi }\right)^2\right)$$ Can Eulers product for $\sin x$ be generalised for any real periodic function with period p and a zero 'a'? If so ...
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0answers
15 views

Periodicity of the fundamental operator of a linear ODE

Consider the homogeneous linear ODE $x'(t)=A(t)x(t)$ where $A$ is $T$-periodic, that is, $A(t+T)=A(t)$ for all $t\geq0$. Let $\Phi$ be the fundamental solution, that is, $\Phi$ satisfies $\Phi'(t)=A(t)...
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1answer
23 views

Problem on continuous periodic functions

Problem. Let $f:\Bbb R\to\Bbb R$ be a continuous periodic function. Show that for every $t>0$, there exists $x\in\Bbb R$ with $$f(x)=\frac{f(x+t)+f(x-t)}{2}.$$ My Attempt. First, if $f$ is ...
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0answers
20 views

Period of a product of periodic functions

I came across the following question: Let $f(x)=\cos(nx) \sin(\frac{5x}{n})$ have a period of $3\pi$. Find the integral value of $n$ The traditional way of solving this is to equate $f(x) with f(x+3π)...
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1answer
36 views

A bound concerning a periodic solution of the Inviscid Burgers' equation

Consisder Inviscid Burgers' equation $$u_t+uu_x=0$$ Assume we are given a smooth solution $u:\mathbb R\times [0,T]\to\mathbb R$ that is periodic in $x$. meaning that for some $K>0$ we have $u(x+K,...
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1answer
31 views

If $\int_x^{x+T}f(t)dt=C$ then $f$ is periodic

If $f$ is a continious function from $\mathbb{R}$ to $\mathbb{R}$ and there exists $C$ such that: $\forall x\in\mathbb{R} :\int_x^{x+T}f(t)dt=C$ then $f$ is $T$-periodic. Proving the ...
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0answers
24 views

Periodic extension for Heat Equation with time-dependent source

I have the PDE $$\frac{\partial u}{\partial t} = \frac{\partial^2u}{\partial x^2} - tu, \quad 0<x<1$$ subject to the boundary conditions (BCs) $u(0,t)=0, u_x(1,t)=0$ and initial condition (IC)...
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0answers
10 views

How to calculate signal periodic and relate to tangent period

How to calculate whether $(-1)^n(-1)^n$ signal is periodic or non-periodic? I know that for a signal to be periodic there has to be such $T$, that satisfies: $$f(n)=f(n+T)$$ What I did: $$(-1)^n(-1)^n=...
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1answer
32 views

Periods of Sine, Cosine and Tangent

I would like to know how to calculate $(-1)^n$ and $(-1)^n(-1)^n$ being periodic and get an explanation for the functions sine, cosine, and tangent periods. I know that for a signal to be periodic ...
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0answers
18 views

Does there exist integer $k >1$ for which $|e^{k\pi}-k\pi-2n|\leq 10^{-6}$

It is well known that $e^{k\pi}-k\pi$ is almost even integer for $k=1$, now ,are there others $k $ ? Assume if there is some finitly $k$ then what about periodicity of $e$ because we would have ...
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4answers
59 views

Functional equation $f(x+1)+f(x-1)=\sqrt{2}\cdot f(x)$

Can $ f: \mathbb{R}\rightarrow \mathbb{R}$ which satisfies functional equation $$f(x+1)+f(x-1)=\sqrt{2}\cdot f(x)$$ be periodic? No idea how to prove this - $f(x+T)=f(x-T)=f(x)...$
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1answer
51 views

General form of a $2\pi$-periodic $C^2$ function satisfying a certain condition

Suppose $f$ is a $2\pi$-periodic $C^2$ function on $\Bbb R$ satisfying $$ \lambda f(x)-\dfrac{\kappa}{2\pi}\int_{-\pi}^\pi \cos (\psi-x)f(\psi)d\psi=\sigma f''(x),$$ where $\lambda, \kappa$ and $\...
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0answers
11 views

keep periodicity when mapping a periodic strip onto another by mapping congruent parts

Let $S$ be a vertical strip-like domain in $\mathbb{C}$ bounded by two simple curves. $S$ is periodic. That means that there is a real constant $c>0$, such that $S = S + ic$ and also $S = S + ic\...
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2answers
58 views

How to show that every non-trivial orbit of ODE is periodic

I'm currently looking at the system of first order ODEs $$ \begin{cases} x' = -y-x^2y \\ y' = x+xy^2 \end{cases} $$ and I try to show that every non-trivial orbit of the system is periodic. I'm ...
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1answer
15 views

Linear rotation spanning 4d

Let $M$ be a real $4 \times 4$ matrix Let $x$ be the time-dependent 4-dimensional vector evolving according to the linear ODE $$ \frac{ d x }{dt} = M x $$ Can you give an example of a matrix $M$ ...
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1answer
51 views

Prove that $\int_a^{a+\pi} |\sin x|dx =$ constant [closed]

Need help proving that no matter what value has "a" (real number) the following integral is always a constant $$ \int_a^{a+\pi} |\sin x|dx = C $$ and whats the value of that constant, thanks
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14 views

A period of LFSR with reducible polynomial

I want to prove the following theorem: Lets assume an $LFSR$(Linear feedback shift register) sequence with a reducible characteristic polynomial of degree n over the finite field $F_q$. Under those ...
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0answers
90 views

Is this relationship between Mandelbrot bulbs a coincidence?

I was messing around with the Mandelbrot boundaries on Desmos and came across something interesting, and I don't have enough experience with the math behind this to conclude whether it's a coincidence ...
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0answers
28 views

Need to show $||S_nf||_2 \leq ||f||_2$ approximation theory

I am considering the inner product space $L_{2 \pi}^2$ of square integrable 2$\pi$-periodic functions on $\Re$, with the inner product defined by: $<f,g>$ = $\int_{0}^{2\pi} f(x)g(x) dx$. For $...
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2answers
62 views

Period of $\sin^{6}(x)+\cos^{4}(x)$

I am trying to find the fundamental period of $\sin^{6}(x)+\cos^{4}(x)$. Now the period of $\sin^{6}(x)$ is $\pi$ and that of $\cos^{4}(x)$ is also $\pi$, so expectedly the period of the expression ...
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3answers
32 views

Every period of function is multiple of fundamental period

Suppose $f : \mathbb{R} \to \mathbb{R}$ is periodic with $T$. Is it necessarily true that $T$ is a multiple of fundamental period $T_0$? Obviously every multiple of $T_0$ is a period. Will the other ...
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1answer
36 views

Why does this functional equation follow from periodicity?

I'm reading a proof where they say that given $$\phi(x) = \Gamma(x)\Gamma(1-x)\sin \pi x$$ and $$g(x) = [\log \phi(x)]''$$ then, since $g$ is periodic with period 1, it satisfies the functional ...
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0answers
49 views

Trying to see if there is correlation between the Corona-Virus transmission rate per day in NY city (COVID-19) and the humidity that day

There should be a correlation between relative humidity and the transmission rate of COVID19. However, since there are no data available for transmission rate per day, I used the # of deaths as a ...
3
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1answer
81 views

Periodic, Infinitely Differentiable Function Dense in $L^2$

I have encountered an interesting question, which seems to have a simple solution. Consider $E$ as the set of $2\pi$ periodic, complex-valued, infinitely differentiable functions s.t. $\forall f\in E$...
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0answers
30 views

Prove Existence Periodic Solution Nonlinear Pendulum with Torque

Suppose we have the following system $$ \theta'=v $$ $$ v'= -bv - \sin\theta + k. $$ I need to prove that there exists a periodic solution in the region where $k>1$. The hint that was given is to ...
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1answer
28 views

How to get the periodic solution of the nonlinear PDE?

How to get the periodic solution of the nonlinear PDE? i.e. the equation $iq_{t} +q_{xx} = i(|q|^{2}q)_{x}$ has the priodic solution $q = ke^{ia[x-(a-k^{2})t]}$, where $a$ and $k$ are ...
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1answer
27 views

Generalization of periodicity

We know that a periodic function (e.g. a trigonometric function) has the property $$ f(x+n\Lambda)=f(x) \qquad n\in\mathbb Z $$ A Bessel function is not exactly periodic, because the value of the ...
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3answers
52 views

Functional equation $f(x+1)=af(x)+b$

Functional equation $f(x+1)=af(x)+b$ There was a question I solved a few days back that asked for a closed form of an equation for a given system. The function came down to this equation which I ...
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2answers
35 views

Find the only periodic solution of an ODE

Find the only periodic solution for $y'+y=b(x)$ with $b:\Bbb{R}\to\Bbb{R}$ has a period of $2T$ and is $1$ for $x (0,T)$ and $-1$ for $x (-T,0)$. The ODE is easy to solve: $y(x) = \exp(-x)\cdot c+1$ ...
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0answers
11 views

Discontinuity of periodic functions with positive and non-summable Fourier coefficients

Let $f : \mathbb{T}^d \rightarrow \mathbb{R}$ be a periodic square integrable function defined over the $d$-dimensional torus $\mathbb{T}^d = [0,2\pi]^d$ with Fourier sequence $(c_n(f))_{n\in \mathbb{...
0
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0answers
21 views

Integral of a periodic function.

Let $f:\mathbb{R}^d\to \mathbb{R}^d$ (for simplicity say $d=3$). Let $\Omega \subset \mathbb{R}^3$. For simplicity assume $f,\Omega$ are as nice (continuous, differentiable, e.t.c) as needed. Let $f$ ...
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1answer
61 views

Proof problem: Show that f is a 3-cycle (redo)

I'm trying to figure out how to solve this problem where, Let $f:R\rightarrow$ $R$ have a cycle {${a_1, a_2, a_3, a_4, a_5}$} where $f(a_i) = a$i+1 , $i= 1,2,3,4$ and $f(a_5) = a_1$. If $...

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