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MathGroup Archive 1993

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Harmonic Function Theory package

  • To: mathgroup at yoda.physics.unc.edu
  • Subject: Harmonic Function Theory package
  • From: xinwei at otter.stanford.edu (Sha Xin Wei)
  • Date: Fri, 9 Apr 93 12:08:13 -0700

[
Sheldon Axler kindly submitted a new edition of the HFT package
to the public ftp archive at otter.stanford.edu.  The documents are
in /ftp/mma/Analysis.

-sxw
Stanford Mathematical Software Users Forum
]


  Date: Thu, 8 Apr 93 17:32:38 EDT
  From: axler at math.msu.edu (Sheldon Axler)
  Subject: read me first

The Stanford math archive includes a Mathematica package (called HFT.m) 

for symbolic manipulation of harmonic functions. (The package is based 

upon material from the book Harmonic Function Theory, by Sheldon Axler, 

Paul Bourdon, and Wade Ramey, published by Springer-Verlag in its 

Graduate Texts in Mathematics series.)
...
The new version of the package uses much faster algorithms for solving 

the Dirichlet problem and other boundary value problems. Users can now 

solve problems in high dimensions with high degree polynomials 

that took too long or too much memory with the previous version. For the 

Dirichlet problem and other boundary value problems, the output may look 

a bit different than in previous versions because the package now 

expresses answers more compactly by using norms.

The package can now solve the Neumann and biDirichlet problems. Given 

polynomials f and g in n variables, the Neumann problem is to find a 

function on the unit ball in R^n whose outward normal derivative on the 

unit sphere equals f and whose Laplacian equals g. The biDirichlet 

problem is to find the biharmonic function on the unit ball of R^n that 

equals f on the unit sphere and whose outward normal derivative on the 

unit sphere equals g (a function is called biharmonic if the Laplacian of 

its Laplacian equals 0.).

For consistency and simplicity, a few of the functions in the package 

have been renamed from the previous version. What was called 

PoissonIntegral is now called Dirichlet (because it solves the Dirichlet 

problem), what was called ExteriorPoissonIntegral is now called 

ExteriorDirichlet. In addition, the two functions in the previous version 

whose names began with Generalized no longer exist; generalized Dirichlet 

problems are now solved by using the same function names used to solve 

standard Dirichlet problems, with an added optional argument used when 

solving a generalized Dirichlet problem (all this is explained in the 

new documentation).

To use the package interactively, simply enter commands as 

described in the new documentation. Users who have written a Mathematica 

package based upon an earlier version of HFT.m should find that their 

packages work fine with the new version of HFT.m if the following 

statements are added to their packages:

     PoissonIntegral = Dirichlet ;
     GeneralizedDirichlet = Dirichlet ;
     ExteriorPoissonIntegral = ExteriorDirichlet ;
     GeneralizedAnnularDirichlet[ f_, g_, r_, R_, h_, x_ ] :=
          AnnularDirichlet[ f, g, h, r, R, x ]

Comments, suggestions, and bug reports about the package, its 

documentation, or the book are welcome. Please send them to me at 

axler at math.msu.edu.

--Sheldon Axler





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