Re: Function of function analytic definition

*To*: mathgroup at smc.vnet.net*Subject*: [mg66226] Re: [mg66148] Function of function analytic definition*From*: "Ingolf Dahl" <ingolf.dahl at telia.com>*Date*: Thu, 4 May 2006 05:21:37 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Narasimham, See http://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula or http://mathworld.wolfram.com/FaadiBrunosFormula.html and The Curious History of Faà di Bruno's Formula (http://www.maa.org/news/monthly217-234.pdf). A Mathematica notebook can be downloaded from MathWorld, http://mathworld.wolfram.com/notebooks/Calculus/FaadiBrunosFormula.nb. Plug this formula into the general Taylor expansion! Hope that helps! Best regards Ingolf Dahl Sweden > -----Original Message----- > From: Narasimham [mailto:mathma18 at hotmail.com] To: mathgroup at smc.vnet.net > Sent: den 1 maj 2006 07:31 > To: mathgroup at smc.vnet.net > Subject: [mg66226] [mg66148] Function of function analytic definition > > Is there perhaps a mixed functional relation between 'n' th > term of Taylor's Series for f(x0 + h) = f '(n-1)(x0)/(n-1)! > and the same for g(x0 + h) = g '(n-1)(x0)/(n-1)! to obtain/ > evaluate 'n' th term of f( g(x) ) and g( f(x) )? Is it not > possible by using Jacobian criteria using multi-parameter > function definitions available on Mathematica to find this > out? [Subject to of course assumption of existence, > uniqueness, smoothness and other well ordered analytic > functional behaviors of f and g]. Hoping for any related comments. > Somehow this comes to me often, earlier queries did not help > clear this up. Regards. >