Determining whether a function is periodic

• To: mathgroup at smc.vnet.net
• Subject: [mg35932] Determining whether a function is periodic
• From: Deirdre Stewart <dstewart at hebel.net>
• Date: Thu, 8 Aug 2002 06:06:22 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Hi,

I've got the feeling I'm overlooking something obvious...

In the framework of an optimization problem, I'm tinkering with a class of
real integral functionals -  i.e. functions defined by an integral - of a real
variable.

I define a functional ff(u) as the integral of a function f(x) with
integration boundaries [0,u]
In Mathematica notation, ff[u] := Integrate[ f(x), {x,0,u} ]

I know that the functional I'm looking for is periodic for some values of u.
My problem is finding the value(s) for x such that ff(u) is perfectly periodic.
The functional ff is quite well-behaved, and uniformly converges towards
a periodic function in u's vicinity.

The human eye is quite good at detecting periods in plots, but surely there's
a clever way to make Mathematica automatically assess a function's
periodicity, given the period value p one expects ?

ff(u) = ff(u + k*p)  where k=integer, p=period

As the period is known, I could e.g. look at the magnitude of the expected
Dirac deltas after calculating a Fourier Transform of the integral functional,
but this seems quite inefficient and prone to numerical errors.

Any suggestions as to how to detect periodicity in Mathematica would be
most welcome...

Deirdre Stewart

```

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