Re: Function of function analytic definition

*To*: mathgroup at smc.vnet.net*Subject*: [mg66172] Re: Function of function analytic definition*From*: dh <dh at metrohm.ch>*Date*: Wed, 3 May 2006 02:44:34 -0400 (EDT)*References*: <e346m0$7r7$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Hi Narassim, I doubt if there is a simple relationship between the n-th term of f1=f[g[x]] and f2=g[f[x]] because in the Taylor Series of f1 and f2 the terms f and g are evaluated for different arguments. Consider f1, then all derivatives of f are avaluated at the point g[x0]. However, for f2, all derivatives of f are evaluated at the point x0. A relationship between these terms must therefore involve all terms of the Taylor Series and is certainly not simple. Daniel Narasimham wrote: > 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. >