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)