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MathGroup Archive 2006

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Re: general form of a n-derivative

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71566] Re: [mg71542] general form of a n-derivative
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Thu, 23 Nov 2006 05:41:37 -0500 (EST)
  • References: <200611221022.FAA04538@smc.vnet.net> <FB3C4201-33DA-4157-B95F-13263EA744C0@mimuw.edu.pl> <6CEACE48-9306-43DC-BD56-55D6188E61C8@mimuw.edu.pl>

There was some confusion involving indexes and powers in my earlier  
posts on this topic so I have decided to restate the result again. Let


f[x_] := Exp[-(1 - x^2)^(-1)]

Then

D[f[x],{x,n}] has the form

(-1)^n*(E^(-1/(1 - x^2))/(x^2 - 1)^(2*n))*p[n][x]

where p[n][x] is a polynomial in x of the form p[n][x]= (n+1)! x^ 
(3n-2) + lower degree terms.

The proof, by induction, is in my two other posts in this thread,  
(after correcting the statement which had the degree of p[x] as 3n+1).

Andrzej Kozlowski
Tokyo, Japan


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