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
votes
3answers
75 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
57 views

Integral $\int_{-\infty}^{\infty}e^{-\frac{u^2}{2 t}}\,\cosh\left(u\right)\,K_{1/2+i\,u}\left(\frac{1}{4x}\right)\,du,\hspace{0.5cm}x,t>0$ is positive

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, ...
11
votes
1answer
336 views

Different way to show $\int_{-\infty}^{\infty}e^{-\frac{\cosh^2(u)}{2x}}\,e^{-\frac{u^2}{2 t}}\,\cos\left(\frac{\pi\,u}{2t }\right)\,\cosh(u)\,du > 0$

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 }\right)\,\cosh(u)\,du,\...
2
votes
1answer
108 views

Prove that $\int_{-\infty}^{\infty}e^{-\frac{\cosh^2(u)}{2x}}\,e^{-\frac{u^2}{2 t}}\,\cos\left(\frac{\pi\,u}{2t }\right)\,\cosh(u)\,du$ is positive

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 }\right)\,\cosh(u)\,...
1
vote
1answer
86 views

In deriving $\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)dx$, why doesn't this imply $f$ is constant?

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 ...
8
votes
4answers
3k views

An integrable and periodic function $f(x)$ satisfies $\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)dx$.

I want to prove: For an integrable function $f(x)$ and periodic with period $T$, for every $a \in \mathbb{R}$, $$\int_{0}^{T}f(x)\;dx=\int_{a}^{a+T}f(x)\;dx.$$ I tried to change the values and ...
9
votes
4answers
8k views

Integral of periodic function over the length of the period is the same everywhere

I am stuck on a question that involves the intergral of a periodic function. The question is phrased as follows: Definition. A function is periodic with period $a$ if $f(x)=f(x+a)$ for all $x$. ...
1
vote
2answers
416 views

Definite integral of a periodic funtion, offset by the period, equals the original definite integral

Suppose $f: \mathbb R \to \mathbb R$ is Riemann integrable on every finite interval and periodic with period $T>0$. Then for every interval $[a,b]$: $$ \int_a^b f = \int_c^d f,$$ where $c = a+T$ ...
1
vote
1answer
41 views

How to find f(23) given part of the graph and that f(x)=f(x+6)? [closed]

I am very confused, how can $f(x+6) = f(x)$ for all $x$ since $f(x+6)$ would be a shift version of $f(x)$, unless it is a straight line? is the answer D? $6+6+6+6 =24$ $\Rightarrow$ $f(23) = f(5) = 4$...
2
votes
2answers
69 views

What is the period of $f(x)=\sin x\cos x$?

Problem We need to find the period of the following: $f(x)=(\sin(x))(\cos(x))$ using basic trigonometric identities which is as follows: My steps disclaimer! I know the steps but I will pin point ...
3
votes
3answers
90 views

Find $\lim_{n\to\infty}\frac{g(t+n)}n$ for $g(t)=\int_0^tf(x)\,dx$, where $f(x+1)=f(x)$

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that $f(x+1)=f(x) \quad \forall x\in\mathbb{R}$. Define $g(t)=\displaystyle\int_0^tf(x)\,dx$, $t\in\mathbb{R}$ and $h(t)=\...
0
votes
2answers
32 views

Integral to periodic function.

I have this question. I would like to help me with this problem please . If $f'(x)$ is a periodic function, with period $a$, prove that $f(x)$ is a periodic function, if and only if $f(a)=f(0)$. I ...
1
vote
0answers
14 views

How is this function a member of $L^{1}(0, \frac{1}{b})$?

The function in question is: $$ F_{n}(x) = \sum_{k \in \mathbb{Z}} f\left(x-\frac{k}{b}\right) g^{\ast}\left(x - na - \frac{k}{b}\right) $$ Where $\ast$ denotes complex conjugation, $f, g \in L^{2}$,...
0
votes
0answers
25 views

Does this modulo-based function have a name?

Consider the scalar periodic function $$ f_{l,u}(x) = ((x - l) \mod (u - l)) + l $$ Where $l$ and $u$ are a lower and upper bound with $l < u$, so that the modulus $u - l$ is always positive and ...
0
votes
0answers
7 views

Intersection of a triangle wave with X-axis

Let's consider a triangle wave given by the formula: $$ y(x) = \frac{4a}{p}(\; |\;(x \; mod \; p) - \frac{p}{2}| -\frac{p}{4}) $$ where $a$ is amplitude and $p$ is period. Problem: how to adjust $...
1
vote
1answer
33 views

periodic period for Markov Chain

I don't understand why the only state with period > 1 is 1 Let's take state 2 for example, what's the period for state 2? Another question is, does an absorbing state(state 4 in this example) only ...
6
votes
3answers
169 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 ...
2
votes
2answers
229 views

Number of zeros in cosine for one period

Stupid I know, but for $ \cos(kx+\phi) $ is the number of zeros in the first period always two? Just unsure of myself.
1
vote
0answers
14 views

Nonlinear first-order differential equation with periodic bounded solution

Let $f:\mathbb R\times \mathbb R\longrightarrow\mathbb R$ be a $C^1$ function, periodic in the first variable, and such that $f(t,1)\leq > 0\leq f(t,0)$ for all $t$. Consider the ...
0
votes
0answers
22 views

Riemann integrability of trigonometric functions clarifying

Consider a function which is periodic with period $2\pi$, and is Riemann integrable on the closed interval $[-2\pi,2\pi]$. Now, can we say that the function is Riemann integrable on every bounded ...
0
votes
0answers
28 views

Projection of periodic trajectories

Let $(\bar x(t),\bar u(t)),\, t\in [0,1]$ solution of $$ \left \{ \begin{array}{l} \dot x_1 (t) = u(t)\, f(x(t)) \\ \dot x_2 (t) = u(t)\\ x_1(0) = 0, x_1(1)=1 \\ x_2(0) = x_2(1) \end{array} \right. $$ ...
1
vote
1answer
36 views

Is this non-constant function periodic for every definable number?

Given the set $\mathbb{D}$ which contains all definable real numbers. The definition must not be infinite long. E.g. it contains $12$, $-3$, $\frac{1}{12}$, $\sqrt{2}$, $\pi^2$, $i+e$, Chaitin's ...
1
vote
1answer
30 views

Integration of periodic function $f \in L^1([0, 2\pi])$

While studying trigonometric series and $L^p$ spaces I was wondering the following: Let's say we have a $2\pi$-periodic function $f \in L^1([0, 2\pi])$ satisfying $\int_{0}^{2\pi}f(x) \, dx = 0$. Is ...
0
votes
1answer
26 views

Singularity of $f(z)=f(z+a)=f(z+b),\ a,b\in\mathbb C$ does not exist

Let $f(z)=f(z+a)=f(z+b),\ a,b\in\mathbb C$ be a not constant meromorphic function, which is periodic and let $a,b$ be lineare independent. Show that $f$ has no zeros or singularities on the boundary ...
1
vote
0answers
14 views

Bound on Rayleigh quotient on $H^{1}_{\text{per}}[0,1]^{n}$

Let $Q=[0,1]^{n}$. Define a $Q$-periodic function as follows: A function $f:\mathbb{R}^{n}\to\mathbb{C}$ is said to be $Q$-periodic if for any $x\in\mathbb{R}^{n}$ and any $z\in\mathbb{Z}^{n}$, it ...
5
votes
2answers
175 views

Integrate a periodic absolute value function

\begin{equation} \int_{0}^t \left|\cos(t)\right|dt = \sin\left(t-\pi\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor\right)+2\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor \end{equation} I ...
0
votes
0answers
24 views

Parabolic-waves train

It can be easily proved that a continuous periodic function of the form: $$f(x)=\mathrm{sgn}\left[\cos\left(\pi x\right)\right]\left[\left(\frac{\arcsin\left[\sin\left(\pi x\right)\right]}{\pi}\right)^...
1
vote
0answers
22 views

How to extend a function to be periodic and smooth?

Assume we have a function f(x) that is twice differentable on [0, L]. Let us define F(x) = f(x) on [0, L], F(x) = -f(-x) on [-L, 0], and F(x + 2L) = F(x) outside of [-L, L]. Thus, F(x) is ...
3
votes
0answers
52 views

Identity theorem for $2\pi\mathrm i$ periodic function

Let $f$ be entire as well as real-valued along the lines $\operatorname{Im}(z)=0$ and $\operatorname{Im}(z)=\pi$. Show that $f$ is $2\pi\mathrm i$ periodic under these circumstances, that is $f(z+2\pi\...
0
votes
0answers
30 views

Inequality between norm of function and it's derivative

There is a theorem: Let $f$ be a continuously differentiable, $2\pi$-periodic function. Given $\int_{-\pi}^{\pi} f(x) dx = 0$, I need to prove that $$||f|| \le \frac{\pi}{2} \cdot ||f'||.$$ Where ...
1
vote
0answers
10 views

Spectral bands of periodic differential operators

I am reading the book "Multidimensional Periodic Schrödinger Operator" (O. Veliev, 2015) which says on page 11: It is well-known the following statements about the spectral properties of $L_{t}(q)$...
0
votes
0answers
45 views

Show periodic extension of $f$

Let $f:]-\frac{\pi}{2},\frac{3\pi}{2}]\rightarrow\mathbb{R}$ where $$f(x)= \left\{ \begin{array}{lcc} x,\ \ \ x\in],-\frac{\pi}{2},\frac{\pi}{2}] \\ \pi-x,\ \ \ x\in]\frac{\...
0
votes
2answers
101 views

What is the period of $f(x) = \cos (x) \cos(2x) \cos(3x)$? [closed]

What is the period of $f(x) = \cos(x) \cos(2x) \cos(3x)$? Please tell me the method plus the logic behind solving these kind of problems .. Plus is there any property for even functions like even ...
6
votes
2answers
126 views

What is the period of $(2007)^{\sin x}$?

What is the period of $(2007)^{\sin x}$? Please explain how to proceed and what's the technique to generally solve these kind of problems.
9
votes
5answers
4k views

Sum of two periodic functions is periodic?

I have following paragraph taken from the Stanford's study material. Question: Is the sum of two periodic functions periodic? Answer: I guess the answer is no if you are Mathematician, yes ...
1
vote
1answer
86 views

For what value of $v_0$ is the solution periodic?

A solution of the second-order differential equation $$ x''+x-x^3=0 $$ satisfies the initial condition $x(0)=0$ and $x'(0)=v_0$. For what value of $v_0$ is the solution periodic? I have tried ...
0
votes
3answers
31 views

Decomposition of periodic functions

Suppose that $f$ is a periodic function defined on the integers with period $mn$, with $m$ and $n$ coprime integers. Does there necessarily exist a function $g$ with period $m$ such that $f-g$ is ...
0
votes
0answers
11 views

Floquet Boundedness for Floquet multiplier $|\lambda_i|=1$

The statement: Consider the system $x'=A(t)x$, where $A(t)$ is a periodic matrix with period T. If $|\lambda_i|=1$ then the corresponding Jordan block to $e^{TR}$ is diagonal. The constant matrix R ...
0
votes
3answers
57 views

If $f(x+1)= -f(x-1)$, prove that the function f is a periodic function and its period is $4$

I sure do know that I should arrive to the point where $f(x)= f(x+T)$. I've tried replacing $x+1=a$ but the problem seems to get more and more complicated.
0
votes
2answers
130 views

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 Spectrum$(S^...
2
votes
3answers
78 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 ...
2
votes
1answer
37 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. ...
0
votes
0answers
53 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
39 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 ...
1
vote
0answers
33 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 ...
0
votes
0answers
30 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)\\ ...
1
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
votes
1answer
98 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
101 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
23 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 ...