From: alex@iastate.edu (Roger Alexander)
Subject: Gaussian moment integral Was: symbolic integral wanted
Summary: Mma bug?
Keywords: Mma, Gaussian moments, symbolic integration
Organization: Iowa State University, Ames IA
Date: Tue, 21 Jul 1992 19:44:12 GMT

In a previous post Richard Waterman (rpw001@fisher.stat.psu.edu)
describes trouble getting Mma to evaluate the integral

=>	ans1 = Integrate[x^a Exp[- b x^2 - c x],{x,0,Infinity}]

and reports

=>	Mma gives an "answer", no problem.

But there *is* a problem, as he recognizes:

=>   If I substitute in some numbers for {a,b,c}, say {2,1.5,-1},
=>   into the symbolic answer I get a numerical result.
=>   
=>   The question is, why isn't the result the same as when I do:
=>   
=>   number2 = NIntegrate:x^2 Exp:- 1.5 x^2 + 1 x:,{x,0,Infinity}:

The answer is, Mma's "answer" to the first problem is wrong.  You 
can check this by looking it up in Gradshteyn/Ryzhik, for example.
Or let Mma display the inconsistency itself:

Mathematica 2.0 for DEC RISC
Copyright 1988-91 Wolfram Research, Inc.
 -- X11 windows graphics initialized -- 

In[1]:= ans1 = Integrate[x^a Exp[- b x^2 - c x],{x,0,Infinity}]

                                                            2
         -1 - a/2                                    a  3  c
        b         c Gamma[1 + a] HypergeometricU[1 + -, -, ---]
                                                     2  2  4 b
Out[1]= -------------------------------------------------------
                                    a
                                 4 2

(* Of course, the integral doesn't even converge unless certain
conditions hold on a,b and c that Mma hasn't bothered to ask us about.
An unsympathetic observer might feel that we deserve what happens
next.  This post merely reports facts; the reader who wishes may
assign blame as seems appropriate. *)

(* We assign values to the constants...*)

In[2]:= subs = {a -> 2, b -> 3/2, c -> -1}

                      3
Out[2]= {a -> 2, b -> -, c -> -1}
                      2

(* ... substitute into the above expression, evaluate numerically ... *)

In[3]:= %1 /. subs

                             3  1
         -HypergeometricU[2, -, -]
                             2  6
Out[3]= -------------------------
                    18

In[4]:= % //N

Out[4]= -0.103054

(* Thus Mma refutes its answer Out[1]: the integrand was positive.  If
we substitute the constants before integrating we get a different
(correct? Exercise!) answer. *)

In[5]:= Integrate[(x^a Exp[- b x^2 - c x])/.subs, {x,0,Infinity}]

                        1/6       1     1                         3  1
        12 + Sqrt[384] E    Gamma[-, 0, -] + 3 HypergeometricU[2, -, -]
                                  2     6                         2  6
Out[5]= ---------------------------------------------------------------
                                      54

In[6]:= % //N

Out[6]= 0.656798

=======================================

I tried Maple V out on this integral too:

    |\^/|      MAPLE V
._|\|   |/|_.  Copyright (c) 1981-1990 by the University of Waterloo.
 \  MAPLE  /   All rights reserved.  MAPLE is a registered trademark of
 <____ ____>   Waterloo Maple Software.
      |        Type ? for help.
> int(x^a * exp( -b*x^2 - c*x ),x=0..infinity);
                        infinity
                           /
                          |       a          2
                          |      x  exp(- b x  - c x) dx
                          |
                         /
                         0

# Maple refuses to evaluate the integral without further information
# on the constants a, b, and c.

> a := 2: b := 3/2: c := -1:
________________________________________________________________________
> int(x^a * exp( -b*x^2 - c*x ),x=0..infinity);
                   1/2           1/2  1/2
            2/27 Pi    exp(1/6) 3    2    + 1/9

                          1/2           1/2  1/2          1/2  1/2
                 + 2/27 Pi    exp(1/6) 3    2    erf(1/6 3    2   )
________________________________________________________________________
> evalf(");
                                  .6567979782
------------------------------------------------------------------------
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