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

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35
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4answers
2k views

Are there periodic functions without a smallest period?

The Wikipedia page for periodic functions states that the smallest positive period $P$ of a function is called the fundamental period of the function (if it exists). I was intrigued by the condition ...
18
votes
2answers
167 views

Is it possible to detect periodicity of an analytic function from its Taylor series coefficients at a point?

Given the Taylor series $\sum a_k (x - x_0)^k$ of an analytic function, it is possible to determine whether the function is periodic more-or-less directly from the coefficients $a_0, a_1, \ldots$ of ...
14
votes
1answer
16k views

Period of the sum/product of two functions

Suppose that period of $f(x)=T$ and period of $g(x)=S$, I am interested what is a period of $f(x) g(x)$? period of $f(x)+g(x)$? What I have tried is to search in internet, and found following link ...
11
votes
2answers
1k views

$\cos(x)+\cos(x\sqrt{2})$ is not periodic

Show that the function $$f(x)=\cos(x)+\cos(x\sqrt{2})$$ is not periodic. I tried $x = a$ and $a\sqrt{2}$. I am guessing that the method of contradiction would be of some help over here. What else ...
11
votes
2answers
283 views

Reversing the process of taking the “sine of an arbitrary shape”

I'm sure we've all seen images such as the following, from wikipedia: link. They give us some nice intuition on what the sine and cosine functions are. Some people may also have seen images such as ...
9
votes
2answers
265 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. ...
9
votes
4answers
242 views

Period of $f(2x+3)+f(2x+7)=2$

I have a problem in finding the period of functions given in the form of functional equations. Q. If $f(x)$ is periodic with period $t$ such that $f(2x+3)+f(2x+7)=2$. Find $t$. ($x\in \mathbb R$) ...
8
votes
2answers
2k views

Is $f(x)=\sin(x^2)$ periodic?

Is the function $f:\Bbb R \rightarrow \Bbb R$ defined as $f(x)=\sin(x^2)$, for all $x\in\Bbb R$, periodic? Here's my attempt to solve this: Let's assume that it is periodic. For a function to ...
8
votes
5answers
1k 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 ...
8
votes
2answers
190 views

Are there functions that have $\Re(f(z))$ periodic but $f(z)$ is not periodic?

Let $f(z)$ be a function meromorphic in a simply connected convex domain $D$ (subset of the complex plane with positive area or the whole complex plane) where $z$ is a complex number. Are there such ...
8
votes
3answers
215 views

Integral inequality on a periodic function

Given $f:\mathbb R\to \mathbb R^+$ continuous and periodic of period $T\geq 0$, I am asked to prove that $$\int_0^T\frac{f(x)}{f(x+\alpha)}\mathrm dx\geq T,\; \forall \alpha\in\mathbb R.$$ How does ...
8
votes
1answer
237 views

Real-analytic periodic $f(z)$ that has more than 50 % of the derivatives positive?

Im looking for a real-analytic function $f(z)$ such that for any $z$ $1) $$f(z+p) =f(z)$ With $p$ a nonzero real number and where $z$ is close to , or onto the real line such that $z$ is in the ...
7
votes
7answers
17k views

How to find period of this periodic function?

How can I find a period of this function? $$2\sin{3x} + 3\sin{2x}$$ Is here any way how to sum both sinuses?
7
votes
3answers
2k views

What is the periodicity of the function $\sin(ax) \cos(bx)$ where $a$ and $b$ are rationals?

So, I have a general question first. What happens to the periodicity when we multiply two periodic trig functions with one another ? The next one is very specific, what is the period of the function ...
7
votes
2answers
422 views

Sturm Liouville with periodic boundary conditions

Background and motivation: I'm given the boundary value problem: $$y''(x)+2y(x)=-f(x)$$ subject $y(0)=y(2\pi)$ and $y \, '(0)=y \, '(2\pi)$. EDIT: These were not given to be zero !! Maybe this ...
7
votes
3answers
248 views

prove a function to be constant

Let the function $f\colon \mathbb{R} \to\mathbb{R}$ have arbitrarily small positive periods, in the sense that if $\delta>0$ then there exists $T \in (0,\delta)$ such that $f(t + T) = f(t)$ for all ...
7
votes
3answers
2k views

Is a broken clock right twice a day?

I was thinking about this expression, and I wondered if it holds true when the clock is slow. I can imagine a slow clock which is not right at all in the span of 12 hours—imagine a clock which ticks 5 ...
7
votes
1answer
149 views

If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?

Let $A$ be a nonzero real number and let $B$ be a nonreal complex number. Let $z$ be a complex number. Let $f(z)$ and $g(z)$ be non-constant functions defined for all complex numbers $z$ and satisfy ...
7
votes
0answers
156 views

number of zeros of complex waves

Does anybody know about any type of methods how to calucalte/estimate the number of the zeros of complex waves (periodic functions as superposition of many harmonic waves) within a given period [0,x] ...
6
votes
3answers
2k views

How do I determine if the following function is periodic?

A question in my textbook asks me to determine if the function $f(x)=\cos(3x)+\sin(x)$ is periodic. I do not believe this to be the case as the arguments of sine and cosine are different and as such ...
6
votes
5answers
518 views

When a periodic function is squared (or cubed, and so on…) does it always lose its periodicity?

For instance $$\sin^{2}\left(-\frac{\pi}{6}\right) = \sin^{2}\left(\frac{\pi}{6}\right)$$ i.e., $\sin^2 (x)$ is an even function and loses the $2\pi$-periodicity of $\sin x $. Is this true in ...
6
votes
4answers
331 views

Prove that f(x)=g(x)

Show that if $f,g:\mathbb{R}\to \mathbb{R}$ are continuous and periodic and $\lim_{x\to \infty}[f(x)-g(x)]=0$, then $f=g$
6
votes
2answers
238 views

determine if a function is periodic

Let $f$ be a continuous and integrable function on $[a,b]$ such that $$\int_a^b f(x)\,\mathrm{d}x = 2$$ and for every $t_1,t_2$ such that $\displaystyle t_2 -t_1 = \frac{b-a}2$ ...
6
votes
2answers
454 views

How to expand the Fourier series for $f(x)=\max \{0, \frac{\pi}{2}-\lvert x\rvert \} $?

My Question: My Goal is to determine the Fourier series for $f(x)=\max \{0, \frac{\pi}{2}-\lvert x\rvert \} \quad$ for $x \in [-\pi, \pi ]$ This function is $2\pi$-periodic. My Approach: i found ...
5
votes
4answers
2k views

Does the phrase “instantaneous frequency” make sense?

I had always thought of time and frequency as being two different (yet complete) descriptions of the same system, so to me, the phrase "instantaneous frequency" didn't make sense -- frequency is a ...
5
votes
1answer
167 views

Prove that a $1$-periodic function $\phi$ is constant if $\phi(\frac{x}{2}) \phi(\frac{x+1}{2})$ is a constant multiple of $\phi$

For $0 < x< \infty$, let $\phi (x)$ be positive and continuously twice differentiable satisfying: (a) $\phi (x+1) = \phi (x)$ (b) $\phi(\frac{x}{2}) \phi(\frac{x+1}{2}) = d\phi(x),$ where ...
5
votes
1answer
425 views

Floquet Theory - Reducing ODE's to Constant Coefficient ODE's?

Am I right in my reading of the bottom of this page by Arnold, where he apparently says that to any system of first order ode's with periodic coefficients one can find a change of variables reducing ...
5
votes
3answers
71 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 ...
5
votes
2answers
626 views

Is the continued fraction of the square root of a base $\phi$ (golden ratio) number periodic when the continued fraction is expressed in base $\phi$?

I have been looking at concise ways to represent irrational numbers using only integers. I was thinking about base $\phi$ (golden ratio base) and how it can represent the quadratic extension of the ...
5
votes
1answer
480 views

Elliptic functions and (meromorphic) simply periodic functions.

Let be $f:\mathbb C\rightarrow\overline {\mathbb C}$ a meromorphic function. The set of periods $\Omega_f$ is a discrete (additive) group and we have one of these possibilities: i) $\Omega_f= \{0\}$ ...
5
votes
2answers
2k views

Proof that a periodic function is bounded and uniformly continuous. [duplicate]

I need to show that if $f:\mathbb{R}\to \mathbb{R}$ is continuous and $\forall x \in \mathbb R, f(x+1)=f(x)$, then: $f$ is bounded, $f$ is uniformly continuous, there exists $c\in \mathbb{R}$ such ...
5
votes
3answers
134 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 ...
5
votes
2answers
208 views

If $f$ is a continuous periodic function with irrational period and if $\sum_n\frac{|f(n)|}{n}<\infty$, then $f$ is identically zero.

Please show that if $f:\mathbb{R}\to\mathbb{R}$ is a continuous periodic function with irrational period and if $\sum_n\frac{|f(n)|}{n}<\infty$, then $f$ is identically zero. (For example, using ...
5
votes
1answer
76 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 ...
5
votes
2answers
143 views

Are these statements of my professor about periodicity of harmonic processes in time series analysis correct?

Assume $X_t$ is a harmonic stochastic process, i.e., $$X_t = \sum_{j=-k}^k A_j \exp(i \lambda_j t)$$ where the frequencies $\lambda_j$ are given and $A_j$ are uncorrelated random variables with zero ...
5
votes
1answer
150 views

Triangular periodic tessellation in two variables

The simple function $f(x, y) = \cos(x) + \cos(y)$ is doubly periodic in a square tessellation. One feature of this surface is that cuts through the nearest extrema in either of two directions (e.g. ...
5
votes
2answers
200 views

For $f$ periodic, $g\to 0$ the integral of $fg$ converges (under some more conditions)

Let $f,g:\mathbb{R}\to\mathbb{R}$ be continuous functions such that: $f$ is periodic, with finitely many zeros in a period The average value of $f$ on a period is $0$ $g$ is monotonic decreasing and ...
5
votes
1answer
111 views

Functions whose $n^{th}$ Derivatives form a cycle

The simplest example of this is $e^x$ which we could say has period 1 (it is its own derivative). $e^{-x}$ would have period 2. Using similar constructions, I can get a function that has a derivative ...
5
votes
2answers
102 views

Integrate a periodic absolute value function [duplicate]

\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 ...
5
votes
1answer
149 views

$x'=Ax$ has one periodic solution. Prove that all solutions are periodic.

I want to prove the following: 1) Suppose $$A_{2,2}=\begin{pmatrix} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end{pmatrix}$$ real and suppose the system of differential equations ...
5
votes
0answers
46 views

If $f$ is $2 \pi$ periodic and $\int_{0}^{2 \pi} f(t) dt=0$ then $\int_{0}^{2 \pi} (f(t))^2 dt \le \int_{0}^{2 \pi} (f'(t))^2 dt$ [duplicate]

Given $f$ a real differentiable function, $2 \pi$ periodic such that $\int_{0}^{2 \pi} f(t) dt=0$ show that: $\int_{0}^{2 \pi} (f(t))^2 dt \le \int_{0}^{2 \pi} (f'(t))^2 dt$. When does equality hold? ...
4
votes
4answers
2k views

Are sin and cos the only continuous and infinitely differentiable periodic functions we have?

Sin and cos are everywhere continuous and infinitely differentiable. Those are nice properties to have. They come from the unit circle. It seems there's no other periodic function that is also ...
4
votes
2answers
464 views

Questions aobut Weierstrass's elliptic functions

from the wikipedia: == In terms of the two periods, Weierstrass's elliptic function is an elliptic function with periods ω1 and ω2 defined as $$\wp(z;\omega_1,\omega_2)=\frac{1}{z^2}+ \sum_{n^2+m^2 ...
4
votes
1answer
68 views

Prove or disprove that $\sin\lfloor x\rfloor$ is periodic.

The title says it all. I was plotting random functions on my phone and noticed this graph. I don't think this function is periodic (WA also agrees). Is there a way to prove if a function like this is ...
4
votes
2answers
600 views

How to show that this real function is not periodic?

How can one prove that $$\cos\left(\frac{\pi}{2} t \right)+\cos\left(t \right)$$ is not periodic? This question is motivated by the harmonic spectral representation of time series. Indeed, it is ...
4
votes
2answers
104 views

prove that $f$ is periodic

A function $f\colon\mathbb{R}\to(0,\infty)$ satisfies equation $f(x)=f(x+64)+f(x+1999)-f(x+2063)$. Prove that $f$ is periodic. I'm quite sure that 1999 and 64 are random numbers (probably 1999 = year ...
4
votes
2answers
109 views

Why is the period of $f$, $\pi$?

I came across a problem, which asked to compute the period of the function $$f(x)=3^{\sec^2x-\tan^2 x}.$$ The answer provided was $\pi$. I don't get how.
4
votes
2answers
119 views

detect largest period in non-harmonic components

let us consider following sinusoidal components $\sin(2\pi 13.5t)+\sin(2\pi 13.99t)+\sin(2\pi 25.3t)+\sin(2\pi 26t)$, clearly this is not periodic in total,because frequencies or periods are not ...
4
votes
2answers
3k views

Period of sum of sinusoids

Say I have a sum of two sinusoids like so: $$ Acos(xt+\phi) + Bcos(yt+\delta) $$ How would I find the period? I know that for just one sinusoid the period would be: $$ Acos(xt+\phi) $$ $$ T = 2\pi/x ...
4
votes
1answer
181 views

Let $f: \mathbb{R} \to \mathbb{R}$ be continuous periodic function with period $T>0$

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous periodic function with period $T>0$. Show that there exists an element $x \in \mathbb{R}$ for which $f(x)=f(x+T/2)$. This is what I came up with ...