From: wrameyxiii@home.remove13.com (Wade Ramey) Subject: Re: Harmonic functions Date: Sun, 14 Jan 2001 23:02:00 GMT Newsgroups: sci.math Summary: What are harmonic functions? The word ³harmonic² is commonly used to describe a quality of sound. Harmonic functions derive their name from a rather roundabout connection they have with one source of sound - a vibrating string. Physicists label the movement of a point on a vibrating string "harmonic motion". Such motion may be modeled using sine and cosine functions, which are sometimes called simple harmonics. The same name is given to the complex exponentials exp(int), n an integer, t real. Now the exponentials are just the restrictions to the unit circle of the homogeneous polynomials z^n and their conjugates. These polynomials all satisfy Laplace's equation. Conversely, any function satisfying Laplace's equation on the disk can be written as a power series in z and its conjugate. So the connection between the solutions to Laplace's equation and the term "harmonic" becomes clear. Historically, however, the term wasn't applied until a similar connection was made in higher dimensions. Analogues of the exponentials exist on the sphere in R^n, n > 2. Again they are the restriction to the sphere of homogeneous polynomials on R^n that satisfy the higher dimensional Laplace's equation. They were given the name "spherical harmonics" by William Thomson (Lord Kelvin) and Peter Tait in 1879. By the early 1900s, the word "harmonic" was applied not only to homogeneous polynomials with zero Laplacian, but to any solution of Laplace¹s equation. Incidentally, your mathematically challenged friends will love the term "spherical harmonics"; some find it almost erotic. ------- Part of this is from a book I coauthored with Sheldon Axler and Paul Bourdon entitled "Harmonic Function Theory". (Second edition should be hot off the presses soon!) Wade ============================================================================== From: Johannes H Andersen Subject: Re: Harmonic functions Date: Sun, 14 Jan 2001 23:47:04 +0000 Newsgroups: sci.math Wade Ramey wrote: [see above --djr] Spherical harmonics are well known in not only quantum physics, but also numerical analysis. A so-called "Spectral model" of the earths atmosphere can be formulated as a truncated series expansion onto spherical harmonics: Derivatives can be performed "exactly" and non-linear advection terms are projected back onto the finite truncation region in a least-squares (Galerkin) fashion using quadrature methods. This is how global weather forecasting is done today. Of interest is stable recursion schemes for finding higher order spherical harmonics. > ------- > Part of this is from a book I coauthored with Sheldon Axler and Paul > Bourdon entitled "Harmonic Function Theory". (Second edition should be hot > off the presses soon!) > Many thanks, I look forward to see this book. Johannes