Re: how to find a function
- To: mathgroup at smc.vnet.net
- Subject: [mg105221] Re: how to find a function
- From: dh <dh at metrohm.com>
- Date: Wed, 25 Nov 2009 02:29:48 -0500 (EST)
- References: <hegehq$17j$1@smc.vnet.net>
barefoot gigantor wrote:
> Dear Math Group:-
>
> Suppose we have:-
>
> x = a * e + (2*b - 3*a^2) * e^2
>
> Now let us find such functions:
>
> F(x) = 1 + a * e + (2 * b - a^2) * e^2 + ......
>
> we are just interested in the first three terms.
>
> Now two such functions can be:
>
> F1(x) = 1 + x + 2 * x^2
>
> and
>
> F2(x) = (1-x)/(1-2*x)
>
> How can we find all such functions F(x)?
>
Hi,
all such function can be written as:
F[x]== 1 + a * e + (2 * b - a^2) * e^2 + e^3 PS[e]
where PS[e] is a power series (terminating or not, converging in a
circle) in e.
To get an expression in x we may solve x for for e:
sol=Solve[x==a * e + (2*b - 3*a^2) * e^2, e]
we may use these solutions above to get F[x] in terms of x. PS[e] can
then be replaced by any analytic function of x.
E.g. one possibility:
F[x]=1 + (a (a - Sqrt[a^2 - 12 a^2 x + 8 b x]))/(
2 (3 a^2 - 2 b)) + ((-a^2 + 2 b) (a - Sqrt[
a^2 - 12 a^2 x + 8 b x])^2)/(4 (3 a^2 - 2 b)^2) + (
fun (a - Sqrt[a^2 - 12 a^2 x + 8 b x])^3)/(8 (3 a^2 - 2 b)^3)
where fun is a function of x.
Daniel