       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

```

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