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