# Talk:Periodic function

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Field: Analysis

## Period of a modified sine graph

How do you find the period of a modified sine graph?

What do you mean? The period is the smallest number a such that
f(x+a) = f(x) for all x
i.e. the point at which it starts to repeat itself
What does your modified sine graph look like?
In general, a sine function $g(x)=a*sin(m*(x+p))+d$ has period $P$ such that $m=(2*Pi)/P$. Note that the coefficient of x in $(x+p)$ must be 1. --anon

## Help wanted at oscillator

There is some content at oscillator that needs sorting out. Please read the proposal at Talk:oscillator Cutler 18:51, 18 Feb 2004 (UTC)

## Overuse of f for function name

The function name f is re-used here, sometimes for an arbitrary function and sometimes for a specific one. Would it be clearer if the function that gives the "fractional part" of its argument were named something else? Frac is a reasonably common name for this operation in programming languages. --FOo 02:57, 9 Dec 2004 (UTC)

## Naturally Occuring Functions

Should this secton contain some real-life periodic functions? i.e tides over a 48 hour perion Ac current etc? --anon

## Constant functions (and one other function)

Should constant functions be considered periodic? If f is a constant function, then f(x+p) = f(x) for all p, in which case there is no “smallest” such positive p.

Also, consider the following function of the real numbers:

$\mbox{f}(x)=\left\{\begin{array}{cl}1&x\mbox{ rational}\\ 0&x\mbox{ irrational}\end{array}\right.$

From

rational + rational = rational
irrational + rational = irrational

it follows that f(x+q) = f(x) for any real number x and rational number q (but again there is no “smallest” such positive q). Should this strange function be considered periodic?

Jane Fairfax 09:58, 12 April 2007 (UTC)

I did not see any requirement on the smallest period in the article. I think these functions are indeed periodic according to the definition. For continuous non-constant functions I think one can prove the existence of the smallest period. For stranger functions, well, we accept them as they are. :) Oleg Alexandrov (talk) 14:49, 12 April 2007 (UTC)

Peter Oliphant: Indeed, I wrote a paper about how periodic functions need not have a smallest period! In fact, I discovered an uncountable number of periodic functions such that they have no smallest period AND they are UNBOUNDED in both the positive and negative directions! This is how. Let I be any irrational number. Then consider the function:

f(x) = a if x = a * I + b [ a,b rational] f(x) = 0 otherwise

This function has every rational as a period, and since 'a' can be as large or as small as desired, the function is unbounded in both the positive and negative directions. THIS is a weird function! Note that the 'otherwise' value need not be '0', but can be ANY real number, and I can be any irrational. Thus, there are an uncountable number of these functions. —Preceding unsigned comment added by 66.166.5.106 (talk) 18:04, 9 February 2009 (UTC)

## Aperiodic

Unfortunately a search on "Aperiodic" links instead to "Periodic", which of course has the opposite meaning and quite different usage. This will cause needless confusion. Could somebody please insert a Disambiguation page and an entry for Aperiodic. Or Whatever is appropriate. Gutta Percha (talk)

The concept of "aperiodic" is essentially the concept of "periodic" seen from the opposite direction:

information about what the one means is also information about what the other means by its negative. I have added an explanation of the word "aperiodic" to the lead of this article, which seems to me the best way of dealing with the issue. JamesBWatson (talk) 09:48, 9 May 2010 (UTC)