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MathGroup Archive 1992

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Integrate 2 March 1992

  • To: mathgroup at yoda.physics.unc.edu
  • Subject: Integrate 2 March 1992
  • From: purvis at mulab.physiol.upenn.edu (bedenbaugh)
  • Date: Mon, 2 Mar 92 20:50:03 EST

Dear Mathgroupers,

Hear's another bug in Integrate in Mathematica 2.0
on a Sun 4/260.

Here is my file "fix.m", copied from an e-mail
message from D. Witthof
supposed to fix 

BeginPackage["Integrate`"]
Clear[HypergeometricURule]
HypergeometricURule =
{
 HypergeometricU[a_,b_,z_. E^(2 I arg[c_/;Znak[c]] + r_.)] :>
 E^(-2 I arg[c] b) HypergeometricU[a,b,z E^r] +
 (1 - E^(-2 I arg[c] b)) Gamma[1-b]/Gamma[1+a-b] Hypergeometric1F1[a,b,z E^r],
  HypergeometricU[1,1,z_/;Znak[z]] :> -E^z ExpIntegralEi[-z],
  HypergeometricU[1,1,z_/;!Znak[z]] :> E^z ExpIntegralE[1,z],
  HypergeometricU[1/2,1/2,z_/;!Znak[z]] :> Sqrt[Pi] E^z Erfc[Sqrt[z]],
  HypergeometricU[a_,b_,z_/;!Znak[z]] :> Pi^(-1/2) E^(z/2) z^(1/2-a) *
      BesselK[a-1/2,z/2]/;Expand[b-2 a]===0
  }

    EndPackage[]


First I load fix.m,
then I try a definite integral.

Process math finished
Mathematica 2.0 for SPARC
Copyright 1988-91 Wolfram Research, Inc.
 -- X11 windows graphics initialized -- 

In[1]:= << fix.m

In[2]:= ans = (1 /(4 Pi^2 k^2)) \
Integrate[(1/(u v)) \
Exp[-((so + ss)/2) (u^2 + v^2) - u v phi], \
{u,-Infinity,Infinity},{v,-Infinity,Infinity}]

Infinity::indet: 
   Indeterminate expression ComplexInfinity + ComplexInfinity encountered.

Integrate::idiv: Integral does not converge.

Out[2]= Indeterminate


The integral does not diverge. It works out via a sequence of
transformations that I would like to teach to Mathematica
(change variable, differentiate, change variable, substitute,
integrate differential equation) (*)

(1/(2 Pi k^2)) ArcSin(phi/(so + ss)) + c

where c is a constant of integration that happens to
be 0.

I know this is not an easy sort of integral to do.
One of the reasons I make the effort to use a 
symbolic math package is so it will save me effort
evaluating hard integrals.  At least, Mathematica
should say it can't do it, not report false information.

This makes me wonder if it safe to try to each Mathematica
the steps.

Any clues/suggestions/workarounds ?

Thanks,

Purvis
purvis at mulab.physiol.upenn.edu

(*) 
see 
R.F. Baum
"The Correlation Function of Smoothly Limited Gaussian Noise"
IRE Transactions on Information Theory
September, 1957

and

S. O. Rice
"Mathematical Analysis of Random Noise"
Bell System Technical Journal, Vol. 23:282-332
and vol 24:46-156, 1945
Reprinted in 
_Selected Papers on Noise and Stochastic Processes_
N. Wax, editor
Dover, New York (1954)





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