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

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21
votes
1answer
27k 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 ...
1
vote
2answers
593 views

Principal period of $\sin\frac{3x}{4}+\cos\frac{2x}{5}$ [duplicate]

Find the principal period of $$\sin\frac{3x}{4}+\cos\frac{2x}{5}$$ It was easy to find principle when single trigonometric function is given, but i don't know how to find principal period of sum of ...
2
votes
1answer
156 views

The difference between two periodic functions converges to zero, is this two functions identical?

If $f(x)$ and $g(x)$ are two periodic functions, that is, $f(x+T_1)=f(x)$ and $g(x+T_2)=g(x)$ for every $x \in R$. Now that $lim_{x\to\infty}(f(x)-g(x))=0$. Conjecture: $f(x) \equiv g(x)$.
8
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 ...
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 ...
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 ...
2
votes
3answers
1k views

Non-trigonometric Continuous Periodic Functions

I've seen lots of examples of periodic functions, but they all have one thing in common: They all involve at least one trigonometric term (e.g. $\sin\theta$, $\cos\theta$, etc.). My question is simple:...
36
votes
4answers
3k 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 ...
8
votes
1answer
240 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 ...
2
votes
2answers
218 views

Prove Parseval Identity for $f \in C(\Bbb T) 2\pi$ periodic continuous functions

Question: Prove Parseval Identity for $f \in C(\Bbb T) $ $2\pi$ periodic continuous functions $$ \frac{1}{2 \pi} \int_{-\pi}^\pi |f(x)|^2 dx =\sum_{n=-\infty}^\infty |\hat f(n)|^2 $$ Thoughts: We ...
3
votes
2answers
3k 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 ...
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 $x$. ...
6
votes
2answers
5k 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 ...
7
votes
2answers
556 views

|f| is periodic implied f is periodic

I can think of an example where this wouldn't hold. Take 1,-1,1,-1,-1,1,-1,-1,-1,1,-1,-1,-1,-1. But I can also prove that the statement holds. Claim: $|f|$ is periodic then $f$ is periodic Proof: ...
6
votes
1answer
83 views

Limit of $\int_E f(nx) dx$ for a $1$-periodic function $f$ on $[0,2\pi]$

Let $E$ be a measurable subset of $[0, 2\pi]$. Assume that $f \in C(\mathbb R)$ is $1$-periodic, i.e. $f(x + 1) = f(x)$. Compute $$\lim_{n\to\infty} \int_{E} f(nx) dx$$. Since $f$ is continuous on $\...
5
votes
4answers
3k 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 ...
2
votes
0answers
74 views

Do there exist periodic fractals $A_f$ of this type?

Let $z$ be a complex number. Meromorphic here means meromorphic on all of the complex plane $C$. Lets define a fractal $A_f$ on the complex plane as the result of iterating a meromorphic function $f(z)...
6
votes
4answers
3k 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 ...
4
votes
2answers
706 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 ...
1
vote
1answer
540 views

Is sin(1/x) periodic?with what period time? [closed]

Is sin(1/x) periodic? with what period time?
1
vote
4answers
224 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 )=\...
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^...
3
votes
1answer
255 views

Minimum period of function such that $f\left(x+\frac{13}{42}\right)+f(x)=f\left(x+\frac{1}{6}\right)+f\left(x+\frac{1}{7}\right) $

Let $ f$ be a function from the set of real numbers $ \mathbb{R}$ into itself such for all $ x \in \mathbb{R},$ we have $ |f(x)| \leq 1,f(x)\neq constant $ and $$f\left(x+\dfrac{13}{42}\right)+f(...
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)\,...
5
votes
2answers
1k views

Period of derivative is the period of the original function

Let $f:I\to\mathbb R$ be a differentiable and periodic function with prime/minimum period $T$ (it is $T$-periodic) that is, $f(x+T) = f(x)$ for all $x\in I$. It is clear that $$ f'(x) = \lim_{h\to 0}\...
3
votes
0answers
87 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 ...
3
votes
1answer
135 views

The periods of the Weierstrass function $\wp(z)$

Is it true that the periods $\omega_1$, $\omega_2$ of $\wp(z)$ are $\omega_1 = 4K$ and $\omega_2 = 4iK'$, respectively? Here, $K = K(k)$ is the complete elliptic integral of the first kind, and $K' = ...
3
votes
2answers
372 views

fundamental period

$f:\Bbb R\to \Bbb R$ is continous and nonconstant function. Let $p$ be a positive real number such that $f(x+p)=f(x) $ for all $x \in \Bbb R$ . Then there exitst $n\in\Bbb N$ such that ${p \over n }=...
3
votes
1answer
49 views

Hidden subgroup problem for $\mathbb{Z} mod 2$

The definition of the Hidden Subgroup Problem (HSP) is as follows (according to a lecture series by Pranab Sen), Let $G$ be a group, $S$ a set and $f : G \to S$ a function. We are given an ...
2
votes
2answers
129 views

Non uniform continuity of a function and almost periodicity

We say that a continuous function $f:\mathbb{R}\to\mathbb{C}$ is almost periodic in the sense of Bohr if: For every sequence $(t'_n)_{n\geq0}$, there's a sub-sequence $(t_n)_{n\geq0}$ such that $f(t+...
2
votes
2answers
35 views

What can be said about a function that is odd (or even) with respect to two distinct points?

This question is a little open-ended, but suppose $f : \mathbb R \to \mathbb R$ is odd with respect to two points; i.e. there exist $x_0$ and $x_1$ (and for simplicity, let's take $x_0 = 0$) such that ...
2
votes
2answers
64 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 ...
1
vote
2answers
70 views

Periodicity of a triginometric function

I have a trigonometric function and I'm interested to know whether or not it has a period. At this stage I'm fairly certain that it is not periodic. However, I don't know how to prove it. Can anyone ...
1
vote
1answer
313 views

How to effectively compute a periodic function?

I'm writing a program to compute a value of periodic function for any arbitrary large argument: $f(k) = (\sum_{n=1}^{2^k} n)\mod\ (10^9 + 7)$, where $n,k \in \mathbb{N} $ I know that $ f(k + P) = f(...
1
vote
1answer
51 views

Periodic Function With A Given Functional Equation

Let $f$ be a real valued function defined for all real numbers $x$ such that for some positive constant '$a$' the equation $f(x+a)=1/2+\sqrt{f(x)-(f(x))^2}$ holds for all $x$. Then prove that $f$ is ...
1
vote
2answers
161 views

Periodic sequence [duplicate]

$(x_n)_n$ is a sequence defined by the relation: $x_{n+2}=|x_{n+1}-x_{n-1}|$ for $n\geq1$ and $x_0,x_1,x_2$ are non-negative integers, not all three equal 0. I think this sequence is periodic, so here ...
0
votes
2answers
129 views

How to determine the orbits of points under the tripling map $f(x)=3x\bmod 1$?

Let $f$ be the tripling map $f(x) = 3x \mod(1)$. Determine the complete orbit of the points $\frac{1}{8}$ and $\frac{1}{72}$. Indicate whether each of these points is periodic, eventually periodic, or ...
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 ...
0
votes
2answers
716 views

Proving the fundamental period of tangent

I'm very new to math and proofs -- so I apologize if my math skills and vocabulary offends you. I have a question that states: Prove that $\pi$ is a fundamental period of the tangent function. I ...
0
votes
0answers
28 views

Single-statement Continuous Periodic function without trigonometry and complex numbers

Superseding the question Periodic function without trigonometry and complex numbers , I am now asking: Can I get a single-statement continuous periodic function without using trigonometric functions ...
0
votes
1answer
100 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 ...
0
votes
0answers
276 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)= \...
0
votes
0answers
11 views

How to determine if a product of periodic functions is periodic? [duplicate]

Consider two periodic functions $f(t)$ and $g(t)$, and suppose that $f(t)$ has a period $T_f$ and $g(t)$ has a period $T_g$. Is $f(t) \cdot g(t)$ necessarily periodic? If so, what is the period? If ...
0
votes
1answer
58 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 : $$\sin\left(\frac{\...
0
votes
1answer
173 views

Function with oscillating frequency?

I'm looking for a function whose frequency oscillates around a certain value (say, oscillating between 440 Hz and 880 Hz, at a rate of 1 Hz -- i.e., its frequency goes up and down once per second, ...
-1
votes
2answers
81 views

Periodic function without trigonometry and complex numbers [closed]

Can I get a periodic function without using trigonometric functions or complex numbers? UPDATE: The question has been superseded by Single-statement Continuous Periodic function without trigonometry ...