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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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