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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1answer
32 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 ...
2
votes
0answers
19 views

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 ...
2
votes
2answers
45 views

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 ...
2
votes
2answers
77 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 ...
0
votes
2answers
26 views

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 $π$ ...
4
votes
0answers
41 views

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 ...
1
vote
2answers
32 views

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 ...
5
votes
3answers
62 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 ...
2
votes
1answer
37 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. $$f(x)=\sum_{t=-\infty}^\infty\frac1{\sqrt{(t+24x)^2+4}}$$ ...
0
votes
2answers
34 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 ...
0
votes
1answer
23 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 ...
9
votes
2answers
239 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. ...
0
votes
0answers
25 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 ...
3
votes
1answer
43 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 $\wp$ - ...
5
votes
1answer
70 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 ...
1
vote
1answer
43 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 ...
0
votes
0answers
36 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 ...
0
votes
1answer
24 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)} = ...
0
votes
0answers
25 views

Show a function is periodic and find the period

Let $x(t)$ be a continuous signal, and $\hat x(u)$ be the fourier transform of $x(t)$. We define $\sigma_T(u)=\frac{1}{T}\sum_{n=-\infty}^{\infty}\hat x(u-\frac{n}{T})$ Show that $\sigma_T$ is ...
2
votes
1answer
26 views

How to find periodic points non-algebraically

For example, $f(x) = x - x^2$ by observation has a fixed point at $x = 0$, and an eventually periodic point at $x = 1$ (that goes to $0$). For the second iteration, to see if there's a periodic point ...
2
votes
1answer
46 views

Same Expansion for Different Functions! What's Wrong?

We know, the function $arctan(x)$ is defined as the real number $y$ such that $tan(y)=x$ and $-\frac{\pi}{2}\leq y\leq\frac{\pi}{2}$ From this definition we can derive the well-known ...
0
votes
0answers
10 views

What is the nomenclature for the repeating part of a curve with n-repeating-peaks?

Below is a Google Trends search for "past papers", notice the curve has repeating portions where each repeat has three peaks at different levels. I want to know what the technical name of such a ...
1
vote
1answer
35 views

It's possible to prove that this integral is positive?

I would like to prove that this function $$F_t(x)=\int_{-\infty}^{\infty}e^{-\frac{u^2}{2 t}}\,\cosh\left(u\right)\,K_{1/2+i\,u}\left(\frac{1}{4x}\right)\,du,\hspace{0.5cm}x,t>0$$ is positive, ...
1
vote
1answer
49 views

The period of a non-linear pendulum

The period of a non-linear pendulum is $T = \sqrt{2} \cdot \int_{-\theta_0}^{\theta_0} \frac{d{\theta}}{\sqrt{\cos(\theta) - \cos(\theta_0)}}$. My problem: what will happened with the period $T$, ...
0
votes
2answers
14 views

Given two piece-wise functions, prove their composition is periodic.

I know how to graph h(f(x)), however, I am having trouble proving it is periodic. i. We must prove that h(f(x + k)) = h(f(x)) for all x ∈ ℝ and for some k ≠0 ∈ ℝ in order to prove that h(f(x)) is ...
2
votes
1answer
51 views

Period of a function?

I am trying to find out the period of a function but this function is giving me a different answer from what I expected: \begin{equation*} f(x) = |\sin x| + |\cos x| . \end{equation*} I know that to ...
-1
votes
1answer
24 views

Complex Fourier Series of $t^3$

I am trying to find compute the complex Fourier series of the following function: $$f(t) = t^3$$ $$-\frac32 \le t \le \frac32$$ $$f(t) = f(t+3)$$ I am using the generic function for the complex ...
0
votes
1answer
22 views

Show that $n\bmod m$ is periodic $\forall n,m \in \mathbb{N}^+$

How can I show that $n\bmod m$ is periodic? If I have a simple example like $n\bmod6 \equiv a$ how can I show that this is periodic if e.g $f(n) = n\bmod6$?
1
vote
1answer
60 views

Evaluating an integral of a periodic function

I have been several days trying to prove that the following integral $$I_t(x)=\int_{-\infty}^{\infty}e^{-\frac{\cosh^2(u)}{2x}}\,e^{-\frac{u^2}{2 t}}\,\cos\left(\frac{\pi\,u}{2t ...
2
votes
1answer
34 views

On the weak closedness of a closed ball with fixed $L^2$-norm in a periodic Sobolev space

Preliminaries: Let $\mathrm{L}_P^2$ denote the Hilbert space of $P$-periodic, locally square-integrable functions $f\colon \mathbb{R} \to \mathbb{C}$ with Fourier series representation $$f(x) \sim ...
2
votes
2answers
67 views

How to prove $ I_t=\int_{0}^{\infty}g(u)\,\cos\bigl(\frac{\pi\,u}{2t }\bigr)\,du$ is positive?

We have the following integral $$I_t=\int_{0}^{\infty}g(u)\,\cos\left(\frac{\pi\,u}{2t }\right)\,du\,\,\,\text{with}\,\,\,\,t>0$$ where $g(u)$ is continuous on $(0,\infty)$, $g(0)=1$ and $g(u)$ is ...
1
vote
0answers
16 views

How to check periodicity of $f(t)$ using samples

Suppose that we know that signal $f(t)$ is $T_1$-periodic. Let $f_1 = 1/T_1$. But we want to know whether signal is $T_2$-periodic also. Let $f_2 = 1/T_2$, and $f_2$ is positive integer multiples of ...
0
votes
1answer
28 views

If $f(t)$ is periodic, is there any $t$ that would equal to DC components?

Suppose $f(t)$ is periodic with period $T$. Would there be $t$ that would necessarily equal to DC component (it can be scaled)? By DC component, I mean $F(0)$ where $F$ is fourier coefficient of $f$. ...
0
votes
2answers
27 views

Simple harmonic motion~

I am stuck in a question which says that: A particle moves on the X- axis according to equation $x=A+Bsin(\omega t)$. The motion is simple harmonic. Find the amplitude of SHM. The answer of the above ...
2
votes
2answers
40 views

How do I find the period of the sine function $y = 20\sin\left[\frac{5 \pi}{2}\left(\frac{x -2}{5}\right)\right] + 100$

Using Desmos I can see the period is $0.8$ but how do I get there? I understand that the period is $2\pi/$co-efficient of $x$ but the $-2/5$ is throwing me off.
5
votes
3answers
125 views

Improper Integral of a periodic function converges

Given $f(x)$ is a periodic function and $\int_0^p{f(x)}dx=0$. Show $\int_1^\infty\frac{f(x)}{x}dx$ converges. 1) I know this integral can be broken into ...
1
vote
2answers
34 views

Periodic Functions f(x)=r(x)/s(x)

I have two questions in one. (1) Let $f(x)=\frac{s(x)}{r(x)}$. If $s(x)$ and $r(x)$ are polynomials of same degree, prove that if $f(x)$ is periodic, it must be constant for all ...
4
votes
1answer
65 views

Behavior of a periodic function

Can a periodic function satisfy $f''(x)f(x)>0, x\in \mathbb{R}$ My intuition says no. Any thoughts on how to approach this?
1
vote
1answer
20 views

Implication of period of derivative

Let $f:(0,1)\rightarrow\mathbb{R}$ be a function with nonconstant derivative $f'$ on $(0,1)$. Is it possible that there exists a real number $t$ such that $f'(x)=f'(x+t)$ for all $x$ such that ...
1
vote
1answer
24 views

prove that $f$ periodic, continuous, $f(x+r)=f(x)+c$ cannot be satisfied with $r\ne 0$ and $c\ne 0$

Prove that if $f$ is periodic, continuous, there are no $r\ne 0$ and $c\ne 0$ constants satisfying $f(x+r)=f(x)+c$ (for all $x$) Let's assume that exist $r$ and $c$ satisfying $f(x+r)=f(x)+c$. Thus, ...
1
vote
1answer
157 views

Frequency of a solution to a PDE

Why is the frequency of $u=\sum_{n=1}^{\infty}B_n$sin$(n\pi x/L)$cos($n^2 \pi^2ct/L)$ equal to the coefficent of $t$ with $L>0,c$ constants, i.e Why is it proportional to $n^2$?
0
votes
1answer
56 views

Can this be a periodic function

$$F(x) = \sqrt{\sin x + \cos x}$$ I graphed this function and it seemed to be periodic but my textbook says that it's not. Is any other rule it's breaking...?
1
vote
0answers
31 views

Existence of periodic orbits (non-linear systems)

I'm trying to solve the following problem: Use the Poincaré-Bendixson's criterion to show that the system has a periodic orbit $$ \dot{x}_1 =x_2 \\ \dot{x}_2=-x_1+x_2-2(x_1+2x_2)x_2^2 $$ The unique ...
2
votes
2answers
74 views

Solution of differential lyapunov equation

How would I solve for following, else any implemented algorithms or solvers in matlab (even ways to solve it) for Lyapunov differential equation of form: $$P'(t) + A(t)^TP(t) + P(t)A(t) + Q(t) = 0,$$ ...
0
votes
1answer
19 views

$U_{n+1} = a * U_n * (1-U_n)$ with : $3<a<3.6$ What are some values of $a$ such that $U_n$ changes its periodicity?

$U_{n+1} = a * U_n * (1-U_n) = a * U_n - a * (U_n)^2$ with : $3<a<3.6$ What are some values of $a$ such that $U_n$ changes its periodicity? I've computed $U_n$ for many different values of ...
0
votes
0answers
20 views

How to write a periodic function expression from a piece of another function

How can I write a periodic function expression from a piece of another defined function?, e.g., how to write a periodic function using the piece of the function $f(x) = x^2$ in the interval $[-L,L]$. ...
0
votes
1answer
17 views

Periodicity of discrete function

Let $x[n]$ a discrete-time signal, $$y[n]= x[2n]$$ I have seen that if $x[n]$ is periodic then $y[n]$ is periodic. Similarly, can we say that if $y[n]$ is periodic then $x[n]$ is periodic as well. ...
0
votes
2answers
84 views

solve integral of $\frac{\sin (ax)}{ \sin(x)}$

I would like to find the area under the curve of $\frac{\sin(ax/2)}{\sin(x/2)}$, namely between the first zero crossing on the left and right: $$ \int_{-\frac{2\pi}{a}}^{\frac{2\pi}{a}} ...
0
votes
1answer
47 views

Modeling with periodic functions

In the month of March, the temperature at the South Pole varies over the day in a periodic way that can be modeled approximately by a trigonometric function. The highest temperature is about ...
1
vote
1answer
61 views

Showing Weierstrass Elliptic Function is meromorphic

Consider the Weierstrass Elliptic function, $$\rho(z) = \frac{1}{z^{2}} + \sum\bigg(\frac{1}{(z-m-nw)^{2}} - \frac{1}{(m+nw)^{2}}\bigg), $$ where $m,n \neq 0,0 $. How can one show that it is ...