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Re: Deleting Integrate[] transformation rule (some progress)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg87680] Re: Deleting Integrate[] transformation rule (some progress)
  • From: UHAP023 at alpha1.rhbnc.ac.uk
  • Date: Tue, 15 Apr 2008 05:54:51 -0400 (EDT)
  • Organization: Low Temperature Physics, RHUL
  • References: <ftq5ab$it$1@smc.vnet.net> <ftscv1$b81$1@smc.vnet.net> <ftv91s$7ur$1@smc.vnet.net>

Dear All,
	Some progress on my original query.  I tried the following where 
the formulae below are expressions which integrate to elliptic 
integrals of the first kind and are from mathematical handbooks.

Unprotect[Integrate];

Integrate[1/Sqrt[1 - m_*Sin[phi_]^2], phi_] := 
  HoldForm[Integrate[1/Sqrt[1 - m*Sin[phi]^2], phi]]

Integrate[1/Sqrt[(1 - v_^2)*(1 - k_^2*v_^2)], v_] := 
  HoldForm[Integrate[1/Sqrt[(1 - v^2)*(1 - k^2*v^2)], v]]

Protect[Integrate];


Subsequent attempts to integrate (eg. [Integrate[1/Sqrt[1 - a*Sin[b]^2], 
b]) do indeed leave the integral undone as required.  However more 
involved expressions such as;

In[52]:=
Integrate[Sqrt[Rvt^2 + (4*R^4)/Rx^4], R] // InputForm

Out[52]//InputForm=
(R*Sqrt[Rvt^2 + (4*R^4)/Rx^4])/3 - 
 (I/3*Sqrt[2]*Rvt^2*Sqrt[Rvt^2 + (4*R^4)/Rx^4]*
   Sqrt[1 - (2*I*R^2)/(Rvt*Rx^2)]*
   Sqrt[1 + (2*I*R^2)/(Rvt*Rx^2)]*Rx^4*
   EllipticF[I*ArcSinh[Sqrt[2]*R*Sqrt[I/(Rvt*Rx^2)]], 
    -1])/(Sqrt[I/(Rvt*Rx^2)]*(4*R^4 + Rvt^2*Rx^4))

still produce the unwanted EllipticF[].  My question is, is this; (a) 
because my above HoldForm[] argument expressions are failing to 
pattern-match the intermediate expressions produced by Integrate[], which 
it then uses to produce results containing calls to EllipticF[], (b) 
because Mathematica  knows more expressions than I entered above
from my handbook which it can integrate up to EllipticF[], or
(c) something else?

Any ideas?

Thanks 
Tom.

Ps. The Email address in the header is just a spam-trap.
-- 
Tom Crane, Dept. Physics, Royal Holloway, University of London, Egham Hill,
Egham, Surrey, TW20 0EX, England. 
Email:  T.Crane at rhul dot ac dot uk
Fax:    +44 (0) 1784 472794


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