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

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Find the period of $|\sin x| + |\cos x - 1|$

I want to find the period of this function , I know that the period of $|\sin x| + |\cos x|$ is $π/2$ but what can a $-1$ do?
4
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1answer
43 views

What is the period of $\sin 2\theta + \sin \frac{\theta}{2}$ [duplicate]

What is the period of $\sin 2\theta + \sin \frac{\theta}{2}$? The period of the first term is $\pi$ and that of the second is $4\pi$. Does that mean that the period of the whole is $4\pi$?
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1answer
31 views

Confused about Fourier series?

From linear algebra we know that if a set of vectors form a basis for a space, their is a unique linear combination of the basis to form any vector in that space. I'm assuming this extends to scalar ...
3
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2answers
59 views

Constructing A Space Filling Curve that fills the Unit Square

I'm reading Neal Carothers' Real Analysis, and he constructs a curve defined over $[0,1]$ that fills the unit square as follows: Let $f$ be a real-valued function over $[0,1]$ such that $f$ is $0$ ...
2
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0answers
35 views

Periodicity of a sum of periodic functions?

The sum of two periodic functions is periodic if: a) Both periodic functions are continuous b) If the ratio of their fundamental periods is rational Can someone explain why the first ...
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1answer
33 views

Derivates of periodic parametric cubic splines

My Problem is sort of solved, I overlooked, that paameters $B$ to $D$ are dependent on $x$ and $y$ one question remains, see bottom of question. I implemented a periodic parametric cubic spline, and ...
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1answer
25 views

Periodic functions proof

I need some help here. Let $f$ be a $2\pi$-periodic function, and define for an arbitrary $k\in\mathbb N$ a function $g(x) = f(kx)$. Show that $g$ is also $2\pi$-periodic. What I've done: $$ g(x) ...
4
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1answer
48 views

Fourier coefficients intuition?

I just learned about Fourier series, and this is how I interpreted them: The complex exponentials form a basis for all periodic functions, and the Fourier series essentially decompose the function ...
3
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1answer
19 views

Is this function/series periodic?

$$f(t)=\sum_{k=-\infty}^{\infty}(-1)^kp_{0.5}(t-2k)$$ Recall: $$p_{\Delta}=\begin{cases}\frac{1}{\Delta},&0\leq t\leq\Delta\\0&\text{ otherwise.}\end{cases}$$ Is the function periodic? If ...
2
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0answers
38 views

When does a linear map become the identify map?

My question is about a linear map defined on the set of smooth periodic functions. Precisely, let $C$ be the set of infinitely many times continuously differentiable and $2\pi$ periodic functions, ...
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1answer
43 views

Dilating sinusoidal period at certain abscissa $x$?

Lets take a look at this function : $$f(x) = \sin\left(\frac{\pi x }{2}\right)$$ when $x$ tends to $1$ this functions get closer to $1$ by bigger values now look at this one : ...
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0answers
27 views

Finding the period of the function

My question is as follows:- Find the trigonometric interpolant $ \bar{f}(x)$ for $f(x)= \frac{\pi}{x+3\pi}$ and $n=1$. Thant is, find coefficients $c_{-1}, c_0 ,c_1$ such that ...
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0answers
32 views

Find what values of 'b' have bounded solution(differential equation)?

$y′′ + b^2{y} = f(t)$ $ f(t) = t$ for $0 < t < 2\pi$ ($2\pi$ periodic sawtooth wave) This is my solution to the differential equation. $y(t) = C_1\cos(bt) + C_2\sin(bt) + b^{-3}\left(bt - ...
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2answers
34 views

Frequency of sinusoidal curve

In this site,The frequency of a trigonometric function is defined as the number of cycles it completes in a given interval. The formula is : frequency=1/period The period of a sine function is ...
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1answer
74 views

Periodic solution to $y\prime = ay + b(x)$

I'm a bit stuck on a certain review question for ODEs. Here's how it goes: Given the one dimensional equation $y\prime(x) = ay+b(x)$, with $a\neq 0$ and $b:\mathbb R\to \mathbb R$ continuous and ...
5
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1answer
152 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 ...
3
votes
2answers
90 views

Understanding the periodicity of a complex exponential function

In the reals, $e^{nx}$ explodes to infinity very fast. But, $e^{inx}$ is bounded and periodic. I am familiar with Euler's formula $e^{ix} = \cos x +i\sin x $. Yet, could you give me some intuition ...
1
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1answer
37 views

Why is a wave with high FM aperiodic?

I was playing with sound synthesis in a program I wrote and I had a wave of the form $\sin(2\cdot\pi\cdot(f_c+\sin(2\cdot\pi\cdot f_m \cdot t)) \cdot t) $ So, just simple frequency modulation. When ...
1
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1answer
28 views

how can we show an antiperiodic function?

How can we graphically show an anti-periodic function? I can't imagine. Maybe I have got no imagination...!! for example we can show the sin and cos or other periodic functions on the graphs. is it ...
2
votes
0answers
30 views

Find the fundamental period

How do I find the fundamental period of this function? $$y = \sin x + \cos(1,01x)$$ I know that the fundamental period of $\sin x$ is $2\pi$ and the fundamental period of $cos(1,01x)$ is ...
3
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1answer
45 views

How many distinct values of floor(N/i) exists for i=1 to N.

Say we have a function $F(i)=\text{floor}(N/i)$. Then how many distinct values of $F(i)$ will exist for all $0 \leq i \leq N$ e.g. We have $N=25$ then. $F(1)=25$ $F(2)=12$ $F(3)=8$ $F(4)=6$ ...
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1answer
32 views

How to determine general solution using Laplace transform?

$y′′ + a^2y = 2u(t-10)$ Here $a > 0$ and is any real number. I am confused by $a^2$ value there. Can anyone show me step by step how to get to up to $Y(s)$? It would really help me understand. ...
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2answers
40 views

How to solve differential equation problem involving Dirac delta function?

$$ y''+2y'+ 10y=b\,δ\left(\, t - T\,\right)\,\qquad y\left(\, 0\,\right)=3\,,\quad y'\left(\, 0\,\right)=0 $$ Can you choose values for $b$ and $T$ ( $b$ and $T$ positive numbers) such that ...
1
vote
1answer
36 views

Differential equation, for which values of 'a' does this have a bounded solution?

Let $f(t) = f(t$) be the 2pi periodic("sawtooth wave"), f(t) = t for $0 \leq t \leq 2\pi$ and consider the equation $$y^{\prime \prime} + a^2y = f$$ For which values of $a$ (here $a$ >0) does this ...
2
votes
1answer
38 views

How to find solution of the integral equation?

$$y(t) + t \int_0^t y(v)dv = 1 + \int_0^t vy(v)dv$$ I found the answer to be $y(t) = \cos{t}$. I have no idea how they go this answer. I would appreciate any suggestions how to solve this.
3
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1answer
43 views

A formal justification for this “physicism”?

I gave a presentation for a seminar class yesterday on Fourier analysis, and introduced the sawtooth function as a counterexample, for a function whose Fourier series is not termwise differentiable. ...
5
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2answers
73 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 ...
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0answers
42 views

period of the sum and the product of sin(ax) and cos(bx)

Take the function f(x)=sin(ax)cos(bx) , with a,b>0 . We know that the fundamental period of sin(ax) is p= 2π/a and the fundamental period of cos(bx) is q=2π/b. Suppose there are positive integer n ...
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1answer
66 views

Period of $\sin(ax)+\cos(bx)$

Take the function $f(x)=\sin(ax)\cos(bx)$, with $a,b>0$. Suppose there are positive integer $p$ and $q$ such that $ap=bq=r$. Then $r$ is a period of $f$ but non always the shortest : for $f(x)= ...
2
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0answers
33 views

Is it Possible to represent $f(x) =\arctan(x)$ as a fourier series ? Why?

Is it Possible to represent $f(x) =\arctan(x)$ as a fourier series ? Why ?
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0answers
28 views

Ode with Piecewise function

We can write this $$12x"+36x'+48x=f(t)$$ my main problem is how to solve this non-homogeneous ODE I know how to do this as 2 different ode unfortunately its not in a syllabus which doesn't use ...
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1answer
34 views

Carifications on periodic functions: periodicity of $f^n, f+g, fg, \frac{f}{g}, f(g)$, constant function, etc.

During a lecture I've been given (only!) the definition of periodic function: Let $A \subset \mathbb{R}, f: A \to \mathbb{R}, t > 0$; $f$ is $t$-periodic iff for every $x \in A$, we have $a+t \in ...
2
votes
2answers
41 views

finding the period of $\sin(2x+3)$

I tried to find the period of $\sin(2x+3)$; looking for $p>0$ such that $ \sin(2(x+p)+3)=\sin(2x+3),$ for all $ x \in R$ which means: $ \sin(2(x+p)+3) - \sin(2x+3)= 0 ,$ for all $ x \in R$ ...
0
votes
1answer
33 views

Laplace transform of a differential equation?

Find the unique solution of $y''+ y = f$, $y(0) = y'(0) = 0$ with the $2\pi$ periodic function given by $f(t)=2\pi \sin(t)$. I am having trouble setting up and starting the the question. I would be ...
1
vote
1answer
44 views

Laplace transform of $f(t)=\left|\sin\frac{t}{2}\right|$?

If you are given a rectified sine wave, $$f(t)=\left|\sin\frac{t}{2}\right|$$ how do you find the Laplace transform of this? I tried using the equation $$L\{ f(t)\} = \frac{1}{1-e^{-sT}} \int_0^T ...
0
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1answer
98 views

Laplace transform of a sawtooth wave

Find the Laplace transform of the periodic function such that $f(t) = t$ if $0\leq t < 2\pi$ I am having trouble setting up this question. Am I on the right path? $$ \mathcal{L}\{f(t)\} = ...
0
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1answer
46 views

How to determine $2\pi$ periodic function?

Let $f(t) = 2\pi \sin t$, and determine a $2\pi$-periodic function $y^∗$ with the property that $\lim_{t\to+\infty} |y(t) − y^∗(t)| = 0$ for every solution $y$ of $y′ + y = f$. I am having trouble ...
0
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3answers
61 views

Differentiability of a function on $\mathbb R$ such that $f(x+1)=f(x)$.

Let, $f:\mathbb R \to \mathbb R$ be a function such that $f(x+1)=f(x)$ for all $x\in \mathbb R$. Then which of the followings are correct? (a) $f$ is bounded. (b) $f$ is bounded if it is continuous. ...
19
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1answer
125 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 ...
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0answers
57 views

Is there exact formula that returns minimal period of a periodic function?

Is there exact formula that returns minimal period of a periodic analytic function? For constant it should return 0, for non-periodic functions - infinity. I only came to the following but it ...
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0answers
44 views

Properties of periodic solutions of nonlinear ODE system

Assume you have a complicated nonlinear ODE system with some parameter $p$. Numerical simulations of the system show, that for any initial conditions and $p$, the solution tends to a periodic function ...
0
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1answer
61 views

Application for interpolating periodic B-spline

I need to draw a cubic C^2 continous, closed (periodic boundary conditions) B-spline which should interpolate a set of control points. If possible it would be great if I could specify the knot vector. ...
0
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1answer
15 views

Proof behind why the multiplication of two discrete time periodic sequences (with the same period) is periodic.

Given two periodic sequences $x[n]$ and $y[n]$ with the same fundamental period $N$, it is intuitive that their multiplication is also a periodic sequence with period $N$. This stems from the fact ...
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0answers
13 views

Order of distribution of the zeros of the interference function of periodic oscillations?

Given a finite (or an infinite) number of periodic oscillations of different shapes but even functions, along the abscissa $x∈R$, every such periodic oscialltion may cross on some zero points the ...
2
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0answers
19 views

Riemann Integrals and Periodic Functions

Consider the following result, where integrals are, say, Riemann integrals: Let $f:\mathbb{R}\to\mathbb{C}$ be a periodic function of period $L$. Then for all $a\in\mathbb{R}$ $$ ...
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1answer
62 views

If $f\in C[0,2\pi]$ and $f(0)=f(2\pi)$ then $f(\theta)=f(\theta+\pi)$ for some $\theta\,\in\,(0,\pi)$?

If $f\in C[0,2\pi]$ and $f(0)=f(2\pi)$ then $f(\theta)=f(\theta+\pi)$ for some $\theta\,\in\,(0,\pi)$? Intuitively I think it's wrong, but I failed to come up with a single counterexample. Can ...
0
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1answer
56 views

On the existence of a positive fundamental period

A function $f$ has period $t$ if for all $x$ in the domain it is true that $f(x+t) = f(x)$. A function is called periodic if it has (at least one) period. Take any periodic function $f$ -periodicity ...
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1answer
147 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 ...
0
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0answers
21 views

Extremum points of Periodic functions

Assume that the function $f,g:\mathbb{R}^n\to\mathbb R$ are $2\pi$-periodic continuous functions (so continuous functions defined on torus $\mathbb T^n$), differentiable almost everywhere in $\mathbb ...
0
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1answer
19 views

Determining Period From A Graph

I'm having trouble understanding what exactly the period is in a graph. My understanding is that the period is horizontal distance between 2 curves in the graph. Like in the following picture which ...