Developing a "custom" special function
- To: mathgroup at smc.vnet.net
- Subject: [mg43958] Developing a "custom" special function
- From: "Alan" <infoNOSPAM at optioncity.net>
- Date: Wed, 15 Oct 2003 04:59:36 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
I have a function defined by a second order differential equation with real coefficients. I'm fairly certain it's not reducible to a special function known by Mathematica. For example, DSolve can't solve it. The differential equation has a two regular singular points: one at zero and one on the real (negative) axis. It has an irregular singular point at infinity. I want to extend this function to the complex z-plane. The reason for that, is that the differential eqn. arises from a Laplace transform. So, I need to do a Laplace inversion using this function. I have managed to extend it along the purely imaginary axis by "smooth pasting" (i.e. analytically continuing) various power series solns in Mathematica. There is no power series about infinity, but there is an asymptotic series there that I also make use of. My methods work, and let me do the inversion, but they have been rather ad hoc. I would like to speed my program up, so I wonder if there is a systematic approach to this type of problem? In other words, I would like to develop a "customized" special function, defined throughout most of the complex plane. Does anyone know any sources of general advice, or have any, on how to approach this problem most efficiently in Mathematica? Thanks so much, alan