Questions tagged [periodic-functions]
Questions on periodic functions, functions $f(x)$ that satisfy the identity $f(x+c)=f(x)$, for some nonzero $c$.
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Why is $|\cos(x)|$ piecewise differentiable on $x \in [-\pi, \pi)$?
I am asked if the $2\pi$-periodic function $f(x) = |\cos(x)|$ is piecewise differentiable on the interval $x \in [-\pi, \pi)$. Plotting the function yields this graph, which shows that there are ...
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Weierstrass elliptic function for quaternions?
What if we generalize and modify the Weierstrass elliptic function for quaternions ?
So our function could have $2,3$ or $4$ periods.
How would that theory be like ?
Is matrix representation the key ...
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Why is this function periodic?
f(x) = {1 if x is rational, 0 if x is irrational}
Why is this function periodic? I tried the following to prove it but couldn't find it satisfying.
What I did was assume that f(x+T) = f(x), with T a ...
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Getting two different periods for the same function
Let
$$f(x)=\frac{1}{2}\left(\frac{|\sin x|}{\cos x}+\frac{|\cos x|}{\sin x}\right)$$
Then find it's period.
My teacher did this question with graphs. He considered the following intervals $0\to\...
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Prove a monotonic function is linear given its "integrals are linear"
This problem is from the MIT Primes 2023 problem set (it's okay to post now):
Let $f : \mathbb R \to \mathbb R$ be a monotonic function. Suppose that $k, l, m, n \in \mathbb R$ with $km \neq 0$ ...
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If $f(x)$ is periodic, can $f(x^2)$ be periodic?
$f(x)$ is a function defined on $\mathbb{R}$, with fundamental period 1. Prove that for all $f$, $g(x) = f(x^2)$ is not periodic, or give a counterexample.
If the condition "fundamental" is ...
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Is the restriction of a periodic function of several variables to a hyperplane also periodic?
Consider a lattice $\Lambda \in \mathbb{R}^n$ and a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ s.t. $\forall x \in \mathbb{R}^n$, $l \in \Lambda$, $f (x + l) = f(x)$. Let $P$ be a hyperplane in $...
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A few questions concerning the Poincare-Bendixson theorem
I have a few questions concerning the Poincare-Bendixson result:($G\subset \mathbb{R}^2$ open, $f\in C_{loc}^{1-}(G,\mathbb{R}^2)$)
Let $u$ be a solution of $u'=f(u)$ so that $\overline{u(\mathbb{R^+})...
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Problem about the relationship of 2 periodic functions
If $f,\varphi$ are two even periodic functions with $T=2$. Suppose
$$f(x) = x(2-x),\quad \varphi(x) = x,\quad x\in [0,1]$$
prove that:
$$f(x)=\sum\limits_{n=0}^\infty\frac{1}{2^{2n}}\varphi(2^nx),\...
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When is a solution $P(f'(x)) = Q(f(x))$ periodic or double periodic?
Consider the differential equation
$$P(f '(x)) = Q(f(x))$$
Where $P(x),Q(x)$ are polynomials.
Examples are $f'(x) = 1 + f(x)^2$ where we get a tan solution and $f'(x)^2 = 4 f(x)^3 - g_2 f(x) - g_3$ ...
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Why Desmos is giving two different answers for integral of the form $\int_{0}^{\pi}\left(\tan^{-1}(\cot(mx)\right)^2\:dx$
Why Desmos is giving two different answers for integral of the form $$I=\int_{0}^{\pi}\left(\tan^{-1}(\cot(mx)\right)^2\:dx,m \in \mathbb{N}$$
Firstly, I used the substitution $mx=\theta$. We get $dx=...
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If $f(x)=\tan^2\left(\dfrac{\pi x}{n^2-5n+8}\right)+\cot(\left(n+m)\pi x\right),\;(n\in N, m\in Q)$ is a periodic function with fundamental Period $2$
If $f(x)=\tan^2\left(\dfrac{\pi x}{n^2-5n+8}\right)+\cot(\left(n+m)\pi x\right),\;(n\in N, m\in Q)$ is a periodic function with its fundamental period as $2$. Then find interval in which $m$ cannot ...
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Derivative of a periodic function with respect to a parameter
Assume we have a real periodic function $f(x)$ with some fundamental period $P$. Let us introduce a coordinate transformation $x=ay$, where $a \in \mathbb{R}$ and $a \neq 0$. Then, on the one hand,
$$
...
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Translation and Scaling on $L^p(\mathbb{T})$
I am trying to understand periodic Lebesgue spaces $L^p(\mathbb{T})$.
In particular, I have trouble understanding how the scaling $[\delta_\lambda f](x) = f(\lambda x)$ and translation $[\tau_h f](x) =...
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Preperiod and Period of the nim-sequence of Octal Games .17 and .117
This refers to a type of impartial game defined as octal games by Berlekamp, Guy and Conway in the first edition of the Winning Ways books.
I noticed that the nim-sequences of $.17$ and $.117$ (first ...
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Any bounded orbit converges to a unique critical point if a system has no periodic orbit
If a system in $\mathbb{R}^2$ has no periodic orbit, and has only one critical point which is asymptotically stable, prove that any bounded orbit $(x(t):t>0)$ must converge to the unique critical ...
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$f(x)= \begin{cases} -1+\sin(k_1\pi x) & x\;\text{is rational} \\ 1+\cos(k_2 \pi x) & x\;\text{is irrational} \end{cases}$
$f(x)= \begin{cases}
-1+\sin(k_1\pi x) & x\;\text{is rational} \\
1+\cos(k_2 \pi x) & x\;\text{is irrational}
\end{cases}$
If $f(x)$ is periodic function, then
(A) Either $k_1, ...
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Is there a general process to find the period of a periodic function? Are there continuous/differentiable non-trigonometric periodic functions?
Is there a general process that I can follow to find the period of a function or to prove that there isn't one?
Everything I find on the matter has something to do with trigonometric functions and ...
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A continuous function whose Fourier series diverges at $0$?
I read this at many places such as wiki or elsewhere
quote :
It is possible to give explicit examples of a continuous function whose Fourier series diverges at $0$.
For instance, the even and $2π$-...
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Proving existence of periodic solutions of a differential equation
As answered in:
Prove that $0$ is the only $2\pi$-periodic solution of $\ddot{x}+3x+x^3=0$
by user Empy2 in comments, to prove that a differential equation has periodic solutions, you must:
Find an $f(...
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3
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Period of $\sin^{2} x\ + \cos^{2} x$
While I was studying Periodicity of functions I encountered the following problem :
My textbook said that :
Constant function is a periodic function but its fundamental period doesn't exist
So to ...
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2
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Entire function taking real values on two parallel lines is periodic [duplicate]
Given an entire function $f$, I have that $f(ix)$ and $f(1+ix)$ are real for all $x \in \mathbb{R}$. I want to show that $f(z) = f(z+2)$ for all $z \in \mathbb{Z}$.
I've thought about considering the ...
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Compute $\lim_n \|u_n-\bar f\|_p$
Let $p \in [1, \infty]$. Let $f \in L^p_{\text{loc}} (\mathbb R)$ be $T$-periodic, i.e., $f(x+T) = f(x)$ a.e. $x \in \mathbb R$. Let
$$
\bar f := \frac{1}{T} \int_0^T f (t) \, dt.
$$
We define a ...
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$u_n \to \overline f$ in the weak topology $\sigma(L^p, L^{p'})$
Let $p \in [1, \infty]$. Let $f \in L^p_{\text{loc}} (\mathbb R)$ be $T$-periodic, i.e., $f(x+T) = f(x)$ a.e. $x \in \mathbb R$. Let
$$
\overline f := \frac{1}{T} \int_0^T f (t) \, dt.
$$
We define a ...
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$2L$ periodic Sobolev space
I am interested in studying the space $H^1([0,1])$. Using Fourier series, I am the. naturally led to studying the corresponding periodic Sobolev space. All references I can find on the internet deal ...
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Fitting a sine function
Consider the following system over a periodic array, where $1\leq i,j\leq n$,
$$
T_j= \sum_{k=0}^n \frac{e^{-\sum_{|i|\leq k}(k-|i|)f_{j+i}/v}-e^{-\sum_{|i|\leq k}(k+1-|i|)f_{j+i}/v}}{\sum_{|i|\leq k}...
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Analitycity of band functions
Let $H=-\dfrac{d^2}{dx^2}+V(x)$, where $V(x)$ is a $2\pi$-periodic and $V\in L^{\infty}(\mathbb R)$.
If $\mathcal H':=L^{2}(0,2\pi)$, we have the decomposition
$\mathcal H=\int_{\mathbb [0,2\pi)}^{\...
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Definition | Cycles And Periodic Functions
I am attempting to come up with a definition for cycles and periodic functions with respects to sequences. I have put the following definitions together, do these look accurate to you? Thanks!
Let $f$...
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Is there a term for a periodic function that has non-discontinuous end behavior?
Basically, a periodic function where the period boundary is $p$
$\lim_{x \to p^-} \left[ f(x) \right]= \lim_{x \to p^+} \left[ f(x) \right]$
As an example, a sine wave would qualify, but not a ...
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Analytical Equation of the gaussian 1D wave equation with periodic Boundary condition
I am trying to validate the 1D analytical wave equation with a numerical solution with periodic boundary conditions. I have implemented the periodic boundary condition for the numerically calculated ...
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Unique $T$-periodic solution to $x'=a(t)x+b(t)$
I think that conceptually I understand this question but I'm struggling to formally it in writing. The questions states "Let $a,b : \mathbb{R} → \mathbb{R}$ be $T$-periodic continuous functions. ...
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On periodic functions like $\sin$ and $\cos$
A problem in Apostol's Calculus [Tom M. Apostol. Calculus, Volume 1, Second Edition (Wiley, 1967). Section 2.19 Exercises,
Problem 21, page 125.]
Suppose the existence of a function $f$ with the ...
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Period $3$ orbit of the logistic map $x_{n+1}=r \cdot x_n(1-x_n)$
One can proof, that the logistic map has an stable orbit of period three for $r=1+2\sqrt{2}$. This can be done by looking at the third iterated of $f$ and investigate it for stable fixed points. For ...
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How to transform a linear system?
Consider the following system
$$
y_i=\sum_{|j-i|\leq k} x_{j}
$$
for some $k< \lfloor (n-1)/2\rfloor$, and with indexes in the $\mathbb{Z}/n\mathbb{Z}$ ring. Essentially, $y_i$ is the sum of $x_i$ ...
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How come the period isn't $\pi$ here?
$f(x)$: $\{x\}+|\cos x|$ and my objective is to find its period. So the period of $\cos x$ is $2\pi$ so then the modulus of $\cos x$ will cause the negative part of the curve of cosine function to ...
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Period of a trigonometric function
Is the function $\frac{\sqrt{\sin(x)}}{\cos(x)}$ periodic? If that is the case, what are the steps to calculate the period?
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If $a_n$ is periodic, show this version $b_n$ with different terms for odd and even indices is periodic too.
Let $a_n = a_1,a_2,a_3,...$ be a periodic sequence and $T$ be its period.
Then the sequence $b_n$ with $b_{2n-1} = a_{(2m+2(n-1)+1)^2+2m+4(n-1)+3}$ for $n = 1,2,3,...$ and $b_{2n} = a_{(2m+2(n-1)+1)^2+...
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If the sequence $a_1,a_2,a_3,…$ is periodic, show that $a_1,a_2,a_9,a_{10}, …=a_{n^2-n+1+(-1)^{n-1}(n-1)}\; , \; n\in \mathbb{N}$ is also periodic
If the sequence $t_i = a_1,a_2,a_3,…$ with $a_i\in \{2,3,5,7\}$ is periodic, show that $s_n = a_1,a_2,a_9,a_{10},a_{25},a_{26},… = a_{n^2-n+1+(-1)^{n-1}(n-1)}\; , \; n\in \mathbb{N}$ is also periodic.
...
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Gradient in toroidal domain
Suppose we have a Gaussian function $$\varphi(x):=e^{-\left(\frac x\sigma\right)^2}\;\;\;\text{for }x\in\mathbb R,$$ $k,d\in\mathbb N$, $x^{(1)},\ldots,x^{(k)}\in[0,1)^d$ and $$f(x):=\sum_{i=1}^k\...
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frequency and periodicity
This is really a basic question:
Let's say we have a message that is sent 2 times each second (periodic message). Thus, we can say that each 0.5s we send a message. But I am trying to understand it in ...
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Fundamental Period vs function continuous atleast once
I know that function is periodic, non constant and continuous at least once $\implies$ function has a fundamental period
My question:
Can we construct a function which is discontinuous everywhere and ...
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Counter-example of continuous Z-periodic functions
In Chapter 16 of Tao's Analysis II, while giving the definition of $C(\mathbb{R/Z},\mathbb{C})$, Tao also has add the following one-sentence claim:
By "continuous" we mean continuous at all ...
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Approximation of the Bernoulli periodic function
I remember seeing a paper that provided a summation approximation of the Bernoulli periodic function which converges when $p\ge 2$;
$$\dfrac{P_{p}(x)}{(p!)}$$
but I don’t quite remember it,
I know for ...
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Is $\sin(440\cdot2\pi x)+\sin(440\cdot 2^{1/4}\cdot2\pi x)+\sin(440\cdot2^{1/2}\cdot2\pi x)$ periodic? [closed]
Can anyone help me to determine whether the function below is periodic or not? If it is periodic, can anyone tell me how to find the period.
$$y=\sin(440\cdot2\pi x)+\sin(440\cdot 2^{1/4}\cdot2\pi x)+\...
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Finding the maximum cycle of a given set
Problem: Given $4$ circles, we define the following set of rules:
i) Any circle which contains $\ge 3 $ elements transfers exactly one of its elements to each of other $3$ circles.
ii) Circles which ...
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Are the mean and variance of a dihedral angle periodic?
In biomolecular science, a dihedral angle is periodic with a period of $2\pi$. It ranges from -180 to 180 degrees. Now, if for 5 dihedral angles, I want to calculate their mean and variance. Will the ...
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Verifying periodicity of a signal/sinusoid in the discrete case
Say I have the signal, $$x(n) = \cos{\left(6.5 n \pi + \frac{\pi}{3}\right)}$$
Periodicity in the discrete case is given by, $\alpha = \frac{2 \pi l}{N}$ where if it is a rational multiple of $2 \pi$ ...
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Combining Multiple Fourier Series
I have two functions that have been modelled with the Fourier Series $f(x)=-x^2$ and $g(x)=-x$, both functions period of $2pi$.
The fourier series of $-x^2$ is given by $5.75+\sum_{n=1}^{\infty}\frac{(...
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Fundamental period of trigonometric equation
The fundamental period of the function $\sin(\frac{\pi[x]}{12})+\tan(\frac{\pi[x]}{3})+\cos(\frac{\pi x}{4})$ is
(A) 12
(B) 24
(C) 36
(D) function is non-periodic
(where $[\cdot]$ represents the ...
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Simplifying expressions such as $e^{-62\pi i/7}$ and $e^{2000\pi i/15}$. Dealing with multiples of $2\pi$ in the argument.
What are the specific steps or rules you should follow when simplifying the argument of a complex number? I am having trouble figuring out the exact methodology when dealing with multiples of $2\pi$ ...
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