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

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14
votes
1answer
15k 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
282 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
124 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
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
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 ...
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 ...
2
votes
2answers
179 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 ...
1
vote
3answers
290 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 ...
0
votes
1answer
56 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 : ...
35
votes
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 ...
2
votes
0answers
69 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 ...
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 ...
4
votes
2answers
440 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
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 ...
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 ...
3
votes
1answer
234 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 ...
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
2answers
641 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 ...
3
votes
0answers
74 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
106 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
1answer
47 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
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
111 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 ...
2
votes
2answers
32 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
306 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 ...
1
vote
1answer
72 views

Showing Weierstrass Elliptic Function is meromorphic

Consider the Weierstrass Elliptic function, $$\rho(z) = \frac{1}{z^{2}} + \sum\bigg(\frac{1}{(z-m-nw)^{2}} - \frac{1}{(m+nw)^{2}}\bigg), $$ where $m,n \neq 0,0 $. How can one show that it is ...
1
vote
2answers
62 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
2answers
128 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 ...
1
vote
1answer
288 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) = ...
0
votes
1answer
117 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
1answer
69 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
2answers
97 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
152 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, ...