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Re: Function of function analytic definition


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.
> 



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