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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