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Saying something good, for a change

Since recent messages of mine were critical of Mathematica and its
documentation, let me share my delight over an instance where
Mathematica's special features enabled a hard calculation to be done
easily.  I need to integrate from 0 to Infinity various linear
combinations of functions of the form

     r^a Exp[-b r^2].   (*)

All these integrals can be done in closed form, and for a variety of
reasons I don't want to use Mathematica's built-in integrator; I must
use the formula.  I programmed it:

gaussParam[alpha_, gamma_] :=
	beta = (alpha+1)/2;
	answer = (1/2) gamma^(-beta) Gamma[beta];

The catch is that the integrand emerges as the output of another
Mathematica function, and I was wondering if I'd need to write a C
program to parse that expression to extract all the a's and b's, or

That same weekend I was reading through Roman Maeder's book,
Programming in Mathematica.  I hit page 70:

     "Often we can avoid programming a loop altogether by applying
     functions to lists or other expressions.  Section[s ...] about
     mapping of functions over expressions go the heart of
     Mathematica's programming language.  Understanding this material
     will allow you to write concise programs in the style that the
     authors of Mathematica think is best to use."

Blinding illumination!  All I need to do is to write a function defined
only on expressions that match the pattern (*), then Map[] it over

gauss[ c_. Exp[-gamma_. r^2] r^alpha_. ] :=
    (c/2) gamma^(-(alpha+1)/2) Gamma[(alpha+1)/2] /; FreeQ[c, r]

gauss[ c_. Exp[-gamma_. r^2]  ] :=
    (c/2) gamma^(-1/2) Gamma[1/2] /; FreeQ[c, r]

gauss[expr_] := Map[gauss, expr]

That doesn't mean that all is sweetness and light, however.  I did run
into two bugs-or-features in this little project, which will be
reported in later messages.

				     Steve Fulling
				     Texas A&M Math

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