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MathGroup Archive 2006

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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.
> 


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