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 and exteriors of balls 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}

The HFT.m Mathematica software package will work on any computer that runs Mathematica. Click below to obtain the appropriate version of the HFT.m software package.

- HFT9.m (11 May 2013; for use with Mathematica version 9)

- HFT7.m (1 January 2012; for use with Mathematica versions 7 and 8)

- HFT6.m (20 December 2008; for use with Mathematica version 6)

- HFT5.m (9 August 2003; for use with Mathematica version 5)

- HFT4.m (7 May 2000; for use with Mathematica version 4)

- HFT3.m (12 June 1999; for use with Mathematica version 3)

- HFT2.m (1 October 1996; for use with Mathematica version 2)

Click below to obtain the appropriate documentation. Starting with Mathematica version 5, the documentation for each HFT.m software package is a Mathematica notebook. For earlier versions of Mathematica, the documentation was a pdf file.

- ComputingWithHarmonicFuctions9.nb (11 May 2013; for use with Mathematica version 9)

- ComputingWithHarmonicFuctions7.nb (11 May 2013; for use with Mathematica versions 7 and 8)

- ComputingWithHarmonicFuctions6.nb (24 December 2008; for use with Mathematica version 6)

- ComputingWithHarmonicFuctions5.nb (20 December 2008; for use with Mathematica version 5)

- ComputingWithHarmonicFuctions4.pdf (15 December 1996; for use with Mathematica versions 2, 3, and 4)

The HFT.m software package is generated by a Mathematica notebook called HFT.nb (except in Mathematica version 2, where the notebook suffix was ma instead of nb). The HFT.nb notebook is not needed to use the HFT.m software package. The HFT.nb notebook is easier for a human to read than the HFT.m software package. Thus the HFT.nb notebook is needed only to examine or change the programming of the HFT.m software package. The HFT.nb notebook and HFT.m software package are linked, so changes made in the HFT.nb notebook will generate a changed HFT.m software package without notifying the user. If you intend to make changes in the HFT.nb notebook, change the name of the notebook first (then changes to the notebook will result in a software package with a new name). Click below to obtain the appropriate HFT.nb notebook.

- HFT9.nb (11 May 2013; for use with Mathematica version 9)

- HFT7.nb (1 January 2012; for use with Mathematica versions 7 and 8)

- HFT6.nb (20 December 2008; for use with Mathematica version 6)

- HFT5.nb (9 August 2003; for use with Mathematica version 5)

- HFT4.nb (7 May 2000; for use with Mathematica version 4)

- HFT3.nb (12 June 1999; for use with Mathematica version 3)

- HFT2.ma (1 October 1996; for use with Mathematica version 2)

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`

).

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.