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

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1answer
27 views

$f$ is smooth and periodic function,$\exists \lambda$ such that $f^{(4)}=\lambda f$ prove:$\exists \lambda$ such that $\lambda= (\frac{2\pi n}{T})^4$

Given $f:\mathbb{R} \rightarrow \mathbb{C}$ is smooth and periodic with period $T>0$ and exists $\lambda \in \mathbb{C}$ such that $f^{(4)}(x)=\lambda f(x)$ for any $x \in \mathbb{R}$. prove: ...
0
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2answers
35 views

Show that an irrationally periodic function is also a constant function [duplicate]

Let $f:\mathbb R \to \mathbb R$ be a function such that for any irrational number $r$, and any real number $x$ we have $f(x)=f(x+r)$. Show that $f$ is a constant function.
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0answers
26 views

On Period of Linear Recurring Sequences modulo $P^e$

If a sequence $ X_0,X_1,X_2,\ldots$ is defined in terms of an initial set $ X_0,X_1,X_2,\ldots ,X_{k-1} $ by the recurrence relation $$ X_{n+k}= ...
2
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0answers
26 views

Examples of periodic functions without sin or cos in their formulations? [duplicate]

I'm trying to find a function f(x) such that NCspline (in Matlab) provides a better interpolation than does the Lagrange polynomial. But this function cannot be a linear combination of the sine and ...
3
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1answer
44 views

Finding period of a recursive sequence defined by modular operator?

If $f(x)$ is defined as : $f(x)=i$ ,if $x\equiv i$ (mod n), $0\leq i< n$ How can I prove whether the following recursive sequences are periodic or not? ...
3
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1answer
67 views

Period of Fibonacci mod $b$?

It is not too difficult to show that the Fibonacci numbers mod $b$ form a periodic sequence. I would like to say something interesting about the period. There is a small shortcut to the brute-force ...
1
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1answer
77 views

Need to prove continuous periodic function of $\varphi (x) \equiv \psi(x)$

Question: Let two $\varphi(x) $ and $\psi(x)$ periodic and continous functions such that $$ \lim_{ x\to\infty}(\varphi(x)-\psi(x))=0, \quad x\in \mathbb{R}. $$ Prove that $$ ...
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2answers
107 views

Finding Period From Functional Equation

How to find the period of real valued function satisfying $f(x)+f(x+4)=f(x+2)+f(x+6)$ ? Note:Use of recurrence relations not allowed.Use of elementary algebraic manipulations is better!
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2answers
99 views

The sum of two continuous periodic functions is periodic if and only if the ratio of their periods is rational?

There are several questions here on MSE about the periodicity of the sum of two continuous, periodic functions. Here's what I know so far: It's obvious that if the ratio of the periods is rational, ...
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5answers
513 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 ...
2
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1answer
67 views

What does $\Bbb R/2\pi$ for a set mean?

I simply cannot figure out what this means. I read this on an article about the scalar product of $2\pi$ periodic functions. it says that < f,g > goes from $\Bbb R/2\pi \to \Bbb C$ (complex) Do ...
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4answers
182 views

Proving a function is continuous and periodic

Suppose we are given a function $$g\left ( x \right )= \sum_{n=1}^{\infty}\frac{\sin \left ( nx \right )}{10^{n}\sin \left ( x \right )},x\neq k\pi , k\in\mathbb{Z}$$ and $$g\left ( k\pi \right ...
1
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0answers
16 views

Why is $\int\limits_{u}^{u+\omega_j} f'(z)/f(z) dz \in 2 \pi i \mathbb{Z}$?

Because $f(u) = f(u+\omega_j)$ for $j \in \{1,2\}$ it applies $$\int\limits_{u}^{u+\omega_j}\frac{f'(z)}{f(z)}dz \quad \in 2\pi i \mathbb{Z} \quad \text{for} \quad j=1,2, $$ Hello, I write my thesis ...
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0answers
17 views

Scalar product with $f(x,y)$ when $ \dot{X} = f(X)$ has periodic orbit

Let $g(x,y), f(x,y) \in C^1: R^2 \to R^2$ such that $f(x,y) \cdot g(x,y)=0$, $\forall (x,y).$ Prove that if $\dot X = f(X)$ has a periodic orbit, then $g$ have a root Intuitively I can see that the ...
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2answers
58 views

Integral of Absolute Value of $\sin(x)$

For the Integral: $\int |\sin (ax)|$, it is fairly simple to take the Laplace transform of the absolute value of sine, treating it as a periodic function. $$\mathcal L(|\sin (ax)|) = ...
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0answers
25 views

about Poincare map

I saw that the Poincare map is defined by the flow of the periodic system with the least period $T$. that is, $$P(x):=\phi_T(x)$$ is a Poincare map with flow $\phi$ of time $T$. but I think if we ...
1
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2answers
49 views

Odd periodic Function

Given an odd function $f$, defined everywhere, periodic with period $2$, and integrable on every interval. Let $g(x) = \displaystyle\int_{0}^{x}f(t)dt$. I know that ...
1
vote
1answer
44 views

Find period of the Function

The given question $$f(x) = \sqrt{\frac{8}{1+x} + \frac{8}{{1-x}}}$$ $$g(x) = \frac{4}{f(\sin x)}+\frac{4}{f(\cos x)}$$ find period of $g(x)$? What I have done putting $\sin(x)$ and $\cos(x)$ in ...
1
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1answer
21 views

Help with Proving : Period estimation for for concatenated sequences

Assuming I have two 8-bit random number sequences $s[n]$ and $d[n]$ which each have a period of $X$ and $Y$ respectively. Therefore: $$s[n+X] = s[n]\\ d[n+Y] = d[n]$$ If they were concatenated ...
3
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2answers
39 views

Defining periodic functions?

Consider $f: (a,a+p] \rightarrow \mathbb{R}$. What is the "formula" for the p-periodic function $g$ which has the property that $g(x + np) = f(x)$ for all $x$ in $(a, a+p]$? I am well aware of how ...
2
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1answer
35 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
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0answers
22 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 ...
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2answers
52 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
83 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
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2answers
32 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
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0answers
46 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 ...
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2answers
36 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
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3answers
66 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
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1answer
41 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
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2answers
44 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
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1answer
26 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
260 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
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0answers
30 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
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1answer
58 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
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 ...
1
vote
1answer
53 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
25 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)} = ...
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0answers
26 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
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1answer
32 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
50 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 ...
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0answers
15 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
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1answer
43 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, ...
0
votes
2answers
19 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
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1answer
57 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 ...
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1answer
26 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
66 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
43 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 ...