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From: master1729 on 26 Apr 2010 10:16
i wrote :

> bobby wrote :
>
> > Could somebody please tell me how to find the
> period
> > of a taylor series ?
> >
> > What is the best way to compute that ?
> >
> > bobby
> >
> >
> > ----
> > This is intresting.
>
> Since nobody wanted to reply or knows the answer ,
> the master will help you out.
>
> let f(x) be a taylor series.
>
> let f(x) be entire and assume it has a period.
>
> then f(x) has the same period as f(x) - f(0).
>
> F(x) = f(x) - f(0) has the fixed point F(0) = 0.
>
> Now do a series reversion :
>
> y = a1 x + a2 x^2 + ...(1)
> x = A1 y + A2 x^2 + ...(2)
>
> plug (2) into (1) and compute the A_n coefficients.
>
> a1 needs to be invertible ( =/= 0 ) but if its not
> invertible redo the above with another fixed point (
> F(a) = a ) or if this doesnt work take the limit a1
> => 0.
>
> Lets call this series reversion of F(x) ; G(x).

some comments :

if you are trying to find the period of an entire function that does not equal zero anywhere , it is an exponential function of another taylor series.

thus you can then take the logaritm.

as long as you avoid x = 0 you can get along with using the G(q) of q = log(x).

by avoid x = 0 i mean that the method below fails for x = 0.


>
> Now the period P is given by - again assuming there
> is one - :
>
> P = +/- [ 2 x - 2 * G( - F(x) ) ]

that isnt clear nor what i meant , * is not intended as product ; ill rewrite :

let A(x) = G( - F(x) )

then period P satifies :

P^2 = [ x - A(A(x)) ]^2

or

P^2 = x^2 - 2 x A(A(x)) + A(A(x))^2


>
> and that P should thus be a constant , otherwise no
> period.
>
> note that a function can have 2 periods as well, like
> e.g. elliptic functions but then it must also have
> poles.
>
> you might or might not be able to find the second
> period based on the above.
>
>
> regards
>
> the master
>
> tommy1729

an example :

period exp(x)

exp(x) has no 0 , thus we can take the logaritm.

the inverse function of x is x.

thus A(x) = log(-exp(x)) = log(-1) + x

P^2 = x^2 - 2 x A(A(x)) + A(A(x))^2

P^2 = x^2 - 2 x (x + 2 log(-1)) + x^2 + 4 x log(-1) + 4 log(-1)^2

P^2 = x^2 - 2 x^2 - 4 x log(-1) + x^2 + 4 x log(-1) ( = 0 ) + 4 log(-1)^2

P = +/- 2 log(-1) = +/- 2 pi i

it works.

tommy1729
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