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

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2
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179 views

Is $a_n = f(a + b c^n)$ when $a_0 = \ln(2)$ and $a_{n+1} = (e+1) a_n (a_n - 1)$?

Definition Let $a_n$ be a real sequence. Assume there exists a continuous real-periodic function $f(x)$ such that $f(n) = a_n$ And $f(x)$ has the period $t$ , where $t$ is An irrational real ...
0
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1answer
31 views

Hot to show that system of nonlinear differential equations doesn't have periodic solutions?

Suppose we have nonlinear system of differential equations $$ \frac{d\mathbf x}{dt} = \hat{A}(\mathbf x, \mu)\mathbf x $$ How to show that it has periodic solutions?
2
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0answers
27 views

Non linear regression calculator

Are there any really good non linear regression calculators around the web? Or is something like matlab the best solution? I tried using excel and its solver tool, but it's complete garbage lol. ...
0
votes
1answer
17 views

Periodic modular piecewise function

For every positive integer $n$, let $\text{mod}_5 (n)$ be the remainder obtained when $n$ is divided by 5. Define a function $f: \{0,1,2,3,\dots\} \times \{0,1,2,3,4\} \to \{0,1,2,3,4\}$ recursively ...
1
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0answers
38 views

Proof for a periodic function

I have to solve the following exercise: The function $f : \mathbb{R} \rightarrow \mathbb{R}$ is a periodic function with $P = 2\pi$ so that $f(x) = f(x + 2\pi)$ is true for all $x \in \mathbb{R}$. ...
0
votes
0answers
22 views

Getting sum of $2 \pi$ periodic function

I have a $2\pi$ periodic function which in the interval $[0,\pi]$ is $f(t) = \sin{\frac{t}{2}}$. I have to find the sum for $t \in \mathbb{R}$. But do I know anything about $f(t)$ outside of $t \in [...
1
vote
1answer
58 views

maximum Difference between two zeros

What is the maximum difference between the two consecutive zeros of the solutions of $y''+(1+x)y=0$ on $0\leq x<+\infty$? I have applied the Strum's comparison theorem (by comparison with $y''+y=0$)...
3
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1answer
44 views

Periodic Poincaré Inequality?

The classical periodic Poincaré inequality states that if $u\in H^1(\mathbb T^n)$ is such that $\displaystyle\int_{\mathbb T^n} u(x)\ dx=0$ then $$\|u\|_{L^2(\mathbb T^n)}^2\leq C_d \|\nabla u\|_{L^2(\...
0
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0answers
17 views

Solve for a Fourier series problem

I am learning Fourier series and I have a problem which has me confused and would like to here others take on it. $$F(t)=\begin{cases}v_0 & \ \ 0 \le t\le T\\ 0 & \ \ T\le t \le \ 2T\end{...
1
vote
1answer
30 views

Let $f(x)=\sin^{23}x-\cos^{22}x$ and $g(x)=1+\frac{1}{2}\arctan|x|$,then the number of values of $x$ in the interval $[-10\pi,20\pi]$

Let $f(x)=\sin^{23}x-\cos^{22}x$ and $g(x)=1+\frac{1}{2}\arctan|x|$,then the number of values of $x$ in the interval $[-10\pi,20\pi]$ satisfying the equation $f(x)=\text{sgn}(g(x)),$ is $(A)6\hspace{...
0
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0answers
71 views

Examples of real orbits with irrational period

Im looking for examples of periodic orbits on the real line given by piecewise analytic functions $g$. More specific at most 2 pieces. Im talking about integer iterations starting at $f(0)=0$ and with ...
1
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0answers
25 views

Floquet Theory : zero equilibirum stability

Wondering if someone could explain exactly what it means when someone says "zero equilibirium is unstable because the floquet multiplier is greater than 1". I understand the FM part but I don't know ...
0
votes
2answers
22 views

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: $$\...
1
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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+...
2
votes
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)-\...
0
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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??...
0
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1answer
98 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 ?
1
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1answer
35 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+...
0
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1answer
41 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 ...
-1
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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 ...
1
vote
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, -...
5
votes
3answers
166 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
votes
1answer
73 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 ...
0
votes
1answer
72 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 ...
0
votes
2answers
58 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 ...
1
vote
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?
1
vote
1answer
55 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 \...
4
votes
3answers
62 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
votes
1answer
83 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 $\...
1
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1answer
45 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 ...
0
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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.
0
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1answer
52 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, ...
0
votes
0answers
50 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
votes
2answers
556 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
votes
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
votes
1answer
97 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 ...
0
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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
7 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 ...
0
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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 ...
2
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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 ...
1
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1answer
85 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 (...
3
votes
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 ...
0
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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 ...
0
votes
0answers
40 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....
0
votes
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
votes
3answers
87 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{$\gamma$}...
1
vote
3answers
47 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
42 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