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

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2answers
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Wave kernel for the circle $\mathbb{S}^1$ - Poisson Summation Formula

Question : How could I compute the (wave) kernel from the fact I have already found (wave) trace on unit circle? The definitions are related to the page $25$ of the following pdf. As the ...
2
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3answers
76 views

Is it true that a cycle with a period of 29 hours over 24 hours leads to a non-recurring pattern and how to prove it?

The default 'reset time' for Internet Information Services is 29 hours. The reason for this is that 'Wade [person on the team who developed the setting] suggested 29 hours for the simple reason ...
3
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1answer
32 views

Periodic function - Differential equation on an existing question

Related to the question : Eigenvalues of the circle over the Laplacian operator, how could I get an explicit formula for the differential equation $g''=\lambda g$ with $g$ a $2 \pi$-periodic function. ...
4
votes
0answers
51 views

Periodic function with the capacity of being $g'' = \lambda g$

Related to the question : Eigenvalues of the circle over the Laplacian operator, what kind of periodic chart $c:(-\pi,\pi)\rightarrow S^1$ has the property that for a continuous function $f$, $g :=f ...
0
votes
1answer
28 views

Differential equations that have non-sinusoidal periodic solutions

Examples help but I mostly just want to know what the criteria is for an equation to give non-sinusoidal periodic functions as solutions. https://en.wikipedia.org/wiki/Periodic_function the first ...
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0answers
24 views

Advanced Trig/Periodic Functions Books?

Before this question is removed for being a duplicate, let me specify what I'm looking for. The book should have a treatment of the foundations of periodic/trigonometric functions. Students are often ...
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0answers
26 views

How to determine this system of ODE's?

I'm facing this problem: "Suppose you have this system of ODE's: $\begin{pmatrix} \dot y (t)\\ \dot x (t) \end{pmatrix} = \begin{pmatrix} a & b\\ c & d \end{pmatrix} \begin{pmatrix} y (t)\\ ...
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vote
1answer
38 views

Passage in a proof from Hofer-Zehnder

The proof I'm referring to is to the following theorem. Assume $S$ is a compact regular and strictly convex energy surface for the Hamiltonian field $X_H$ in $\mathbb{R}^{2n}$. Then $S$ carries a ...
4
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1answer
94 views

Find the number of points of distance n away from origin as function of n

I came across a seemingly simple problem the other day and I thought I'd share it with anyone interested. Say you have a point in 3 dimensions. The number of points that are of distance $0$ away is ...
3
votes
2answers
96 views

Why $\sin(x)+\sin(\pi x)$ is not periodic?

Why $\sin(x)+\sin(\pi x)$ is not periodic? There is an answer here which tries to explain it, but I somehow do not get it. If we assume that $T>0$ is a period of $\sin(x)+\sin(\pi x)$, then ...
2
votes
0answers
20 views

Estimating pseudo-periodicity of signals

I have pressure data which are measured at a given point in a standing wave. These data(signals) are 'almost' sinusoidal in nature. Each cycle may slightly vary from the original signal i.e the ...
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votes
3answers
87 views

How to find a formula for a periodic sequence?

I would like to find the formula for a periodic sequence such as 4, 1, 1/4, 1/4, 1, 4... with a period of 6
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1answer
64 views

Question about a continuous periodic function [closed]

Consider the continuous and periodic function $f:\mathbb R \rightarrow \mathbb R$ with period $T > 0$ so that $f(x)=f(x+T)$ for any $x$. Question: Prove that there exists a $c$ such that ...
0
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0answers
12 views

periodic function sine wave problem relate with sound

SinD +SinC = 2Sin (C+D)/2 Cos(C-D)/2 can anyone help me on how to express this equation as single wave in the case that two sine waves those sum makes a sound? Thank You!
3
votes
3answers
44 views

Fundamental Period of $\tan x \cot x$

What is the period of $\tan x \cot x?$ I was given this question today. What I did was simplify the expression , and it reduced to a constant function. So it had no fundamental period. But my teacher ...
0
votes
1answer
6 views

How is Dulac's Multiplier selected?

I'm aware that Dulac's (Negative) Criterion states that a system of differential equations of the form $x' = f(x,y), \; y' = g(x,y)$ has no periodic orbits in the plane if we can find some function ...
1
vote
1answer
34 views

period of the function $y = \sin^2\frac{\sqrt2x+3}{6\pi}$

Today I came across a question The fundamental period of the function $y = \sin^2\frac{\sqrt2x+3}{6\pi}$ is $\lambda \pi^2$ then the value of $\frac{\lambda}{\sqrt2}$ is ___ I tried to equate ...
2
votes
0answers
85 views

Periodic solution to DDE: $\frac{d}{dt}x(t)+\left(\frac{\pi}{2}+\epsilon\right)x(t-1)[1+x(t)]=0$

Consider differential equation with delay: $$\frac{d}{dt}x(t)+\left(\frac{\pi}{2}+\epsilon\right)x(t-1)[1+x(t)]=0.$$ Let's use $t=(1+c)\tau$ substitution to normalize time t: ...
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0answers
24 views

How to determine the smallest period of a parametric curve?

Consider the polar function $r(\theta) = \sin(3\theta)$, and the parameterization of its graph given by $x = \sin(3\theta)\cos(\theta), \;y=\sin(3\theta)\sin(\theta)$. Upon inspection, one can observe ...
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0answers
11 views

How to determine if a product of periodic functions is periodic? [duplicate]

Consider two periodic functions $f(t)$ and $g(t)$, and suppose that $f(t)$ has a period $T_f$ and $g(t)$ has a period $T_g$. Is $f(t) \cdot g(t)$ necessarily periodic? If so, what is the period? If ...
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0answers
19 views

How to find the fundamental period of the product of two periodic functions? [duplicate]

Say you wanted to find the period of $x(t)=\sin(at)\sin(bt)$ where $a$ and $b$ are real constants. How could you go about this?
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2answers
21 views

Fourier series: can a function be odd and have a dc component?

Long story short: fourier series is taken in two subjects (for now). One doc says that the dc component is 0 if the function is odd. The other says that odd and even has no effect on the dc ...
1
vote
1answer
58 views

Integration of periodic functions

I have a question at hand (which may be easy to some) ,but unfortunately I don't know how to even begin with. Could someone help me? If $f$ and $g$ are continuous, $2\pi$ periodic functions then ...
11
votes
1answer
318 views

How to prove that a function is positive

I have been long time 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 ...
0
votes
1answer
18 views

Construct a Sinusoidal Equation for an Irregular Period

I would like to be able to construct a sinusoidal function of limited domain given a set of real roots, assuming that the function is graphically centered on $y=0$. I expect that this would ...
1
vote
1answer
50 views

What is the period of a fuction which satisfies the condition $f(a-x)=f(a+x)$?

What is the period of a function which satisfies the condition $f(a-x)=f(a+x)$ where a is any positive integer? I tried substituting $x$ with $x-a$ but that does not seems to help me a lot. I ended ...
2
votes
2answers
3k views

Finding the period of complex exponential function

I am having some trouble finding the period of the following discrete signal: $x[n]=e^{jn2\pi/3}+e^{jn3\pi/4}$
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1answer
31 views

My brain doesn't work right now: What's the formula for the $n$th vertex of a discretized sine wave?

So far I have: $$ A \sin(2\pi f ? + \phi) $$ where $f$ is cycles per second, and $\phi$ is in seconds. If I'd like to approximate the sine wave with $N$ points per cycle, and I want to draw $C$ ...
2
votes
1answer
38 views

Is there a family of functions that includes triangle, sin, and square waves?

Is there a family of functions that includes triangle, sin, and square waves? ]2 If so, is there a way to parametrise them such that a single parameter sweeps from triangle through sin to square? ...
0
votes
1answer
23 views

Periodic Foricing Terms

The question asks to find the solution for the initial value problem: $ y''+\omega^2y=sin(nt),\quad y(0)=0,\quad y'(0)=0 $ where $n$ is a positive integer when a) $\omega^2\neq n^2$ and b) ...
1
vote
1answer
27 views

Provinf Orbits are eventually fixed or eventually prime-2-periodic

Please I need help with this question: Let $S = \{a,b,c,d\}$ be a finite set and suppose that $f \colon S \to S$ has the property that: $$f(a) = f(b) = f(c) = d$$ Prove that each of the orbits of ...
0
votes
0answers
16 views

What is the periodicity of an infinite sum of Dirac delta functions?

I have the following function: $$F(\mu)=\sum_{n=-\infty}^\infty [\delta(\mu - n/\Delta T - a) + \delta(\mu - n/\Delta T + a)]$$ Where $\Delta T$ and $a$ are positive real constants. It is a result ...
1
vote
1answer
19 views

Sum of a periodic sequence of functions

Suppose that $x_j$ is an $n$-periodic sequence. Show that $$\sum_{j=m}^{m+n-1}x_j=\sum_{j=0}^{n-1}x_j.$$ So far I have tried playing around with the indices of the sequence and have \begin{align*} ...
2
votes
1answer
102 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
37 views

problem in Functions and Periodicity

What is the period of $f(x+\frac{1}{2}) +f(x-\frac{1}{2})=f(x)$ ? I tried substituting $x=x+\frac{1}{2}$ and $x=x-\frac{1}{2}$ but that didn't get me anywhere. According to the standard procedures , ...
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1answer
27 views

show that an ordinary differential equation has T periodic solution

I have $dx/dt=-x^3(t)+h(t)$ where h(t) is a smooth, T-periodic function. Show that $x'(t)$ has a periodic solution. So I tried solving the function as letting $h(t) =sin(t)$ and $cos(t)$ which are ...
2
votes
1answer
55 views

Layperson's explanation of Euler's formula

A few weeks back I asked a question which lead to Euler's formula being brought up. I don't have the mathematical background to fully appreciate it's purported mathematical beauty. Just yesterday I ...
4
votes
1answer
109 views

Is $(\sin{x})(\sin{\pi x})$ periodic?

I'm wondering, is the function $f=(\sin{x})(\sin{\pi x})$ is periodic? My first inclination would be two assume that if the periods of the individual sine expressions, $p_1 \text{and}\space p_2$ have ...
6
votes
1answer
78 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 ...
0
votes
2answers
27 views

Can't get the period of the sum of a sine and a cosine

$$ x(t) = 2\cos(5t+\pi/10) + 3\sin(5\pi t) $$ I'm supposing that the signal is periodic (because sine and cosine are periodic) but then; \begin{align} P_{\sin} &= \tfrac{2}{5} \\ P_{\cos} &= ...
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0answers
26 views

Finding the period from a graph to create a function

I'm trying to convert a graph from this website into a function where y = d + A sin (k (x - delta) ) I have already gotten the vertical shift, amplitude and the ...
2
votes
3answers
47 views

Constant functions periodic?

I dont understand the meaning of this line in my book - " $\sin^2x + \cos^2x$ is periodic but the fundamental period is not defined. " Why is the period not defined? $F(x)$ is $1$ here so it is a ...
1
vote
1answer
83 views

How to find periodicity of a nim sequence?

I am trying to solve a problem which is a simple algorithmic game. Link to the problem - https://community.topcoder.com/stat?c=problem_statement&pm=6856 I have basically figured out that for ...
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vote
1answer
43 views

how to find its period?

I have this function $T(x)=x+4B \pmod A$. I want to solve the congruence for the smallest positive $n$, $T^{n} (x)=x \pmod A$. How to solve it and find its period? To solve it, what I did is by ...
0
votes
1answer
26 views

Period of a trigonometric series

What is the period of the function represented by the series $a_1 cos x+a_2cos2x+a_3cos3x+...$ I guess it is $2\pi$. Am I right?
1
vote
2answers
50 views

Laplace Transform of Square Wave Function

I am given a problem in my textbook and I am left to determine the Laplace transform of a function given its graph (see the attached photo) - a square wave - using the theorem that $$F(s) = ...
6
votes
1answer
62 views

When does a linear combination of trigonometric functions have an axis of symmetry?

I am trying to find out when a linear combination of $\sin(ax)$ and $\cos(bx)$ has an axis of symmetry. Clearly, $\sin(x)+\cos(x)$ has an axis of symmetry at $\pi/4$. It seems as if $\sin(3 ...
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1answer
66 views

How to prove that the ellipse is a periodic orbit knowing that the orbital derivative of a function V is zero on there

The question is as follows: Show that the orbital derivative of the function $V=(1-x^2-2y^2)^2$ is zero on the ellipse $x^2+2y^2=1$, and explain why you can deduce that the ellipse is a periodic ...
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0answers
43 views

Periodicity of a trigonometric function

How to find the fundamental period of the function $|\sin x - \cos x|$ and $|\sin x - \cos x|$ + $|\sin x + \cos x|$? Please tell the proper method to find the period of the functions like these. I ...
0
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0answers
55 views

about Poincare map

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