The HFT.m Mathematica software package performs symbolic manipulation of expressions that arise in the study of harmonic functions. This software, which is available electronically without charge, can perform symbolic calculations that would take a prohibitive amount of time if done without a computer. For example, the Poisson integral of any polynomial can be computed exactly.

Some of the capabilities of this software:

- symbolic calculus in
**R**^{n} - Dirichlet problem for balls, quadratic regions, annular regions, and exteriors of balls in
**R**^{n} - Neumann problem for balls, exteriors of balls, and ellipsoids in
**R**^{n} - biDirichlet problem for balls in
**R**^{n} - the Bergman projection problem for balls in
**R**^{n} - finding bases for spherical harmonics in
**R**^{n} - explicit formulas for zonal harmonics in
**R**^{n} - manipulations with the Kelvin transform
- Schwarz functions for balls in
**R**^{n} - harmonic conjugation in
**R**^{2}

Click below to obtain the current version of the HFT.m software package:

- HFT12.m (version 12.03, 20 December 2020; for use with Mathematica versions 7 through 12)

Click below to obtain the current documentation as a Mathematica notebook:

- ComputingWithHarmonicFunctions12.nb (20 December 2020; for use with Mathematica versions 7 through 12)

Click below to obtain the current documentation as a pdf file:

- ComputingWithHarmonicFunctions12.pdf (20 December 2020; for use with Mathematica versions 7 through 12)

Click below to obtain the Mathematica notebook that generates the HFT12.m package. The HFT12.nb notebook is not needed to use the HFT12.m software package. However, the HFT12.nb notebook is easier for a human to read than the HFT12.m software package. Thus the HFT12.nb notebook should be used to examine or change the programming of the HFT12.m software package. The HFT12.nb notebook and HFT12.m software package are linked, so changes made in the HFT12.nb notebook will generate a changed HFT12.m software package without notifying the user. If you intend to make changes in the HFT12.nb notebook, change the name of the notebook first (then changes to the notebook will result in a software package with a new name).

- HFT12.nb (version 12.03, 20 December 2020; for use with Mathematica versions 7 through 12)

All items linked above are copyrighted by Sheldon Axler but are distributed without charge.

Please send suggestions for additional features and error reports to Sheldon Axler (`axler@sfsu.edu`

).

To request the versions of this package that work with versions of Mathematica earlier than version 7, contact Sheldon Axler (`axler@sfsu.edu`

).

Many of the algorithms used by this software are based on material in the book listed below, published by Springer in its Graduate Texts in Mathematics series. The software can be used without the book, just as the book can be used without the software. Click below to learn more about the book.

Some of the algorithms used by this software are explained in the paper listed below. Click below to learn more about this paper.

Some of the algorithms used by this software are explained in the paper listed below. Click below to learn more about this paper.

Some of the algorithms used by this software are explained in the paper listed below. Click below to learn more about this paper.