From: harper@cayley.mcs.vuw.ac.nz (John Harper) Subject: Re: hypergeometric functions Date: 5 Apr 2000 21:57:23 GMT Newsgroups: sci.math.symbolic Summary: [missing] In article <050420001522173962%edgar@math.ohio-state.edu.nospam>, G. A. Edgar wrote: >In article , Frederic van Wijland > wrote: > >> I am looking for the asymptotic behavior of the hypergeometric >> function 1F2(a;b,c;z) for z--> +infinity (I have a=2, b=3/4 and >> c=5/4). There is no clear account in Abramowitz nor in Gradstein. >> Thanks in advance to those who can provide me with some help. > >Maple doesn't provide an asymptitic expansion either, but >it does do this... It's a pity Maple V.5 didnt supply these (I had complained to Maple about their absence from V.4; I don't know whether they made it to V.6). Whenever I have a special function problem that Maple, Abramowitz&Stegun, and Gradshteyn&Ryzhik all fail to solve, I try the following: Prudnikov A P, Brychkov Yu A & Marichev O I Integrals and Series (5 vols Gordon & Breach vol 1 1986 sorry I don't have dates of the others) Luke Y L The special functions and their approximations (2 vols Academic Press 1969) Wright E M (unlike the above books, these papers are just on hypg and related functions): Journ Lond Math Soc 10, 286-293, 1935; Proc Lond Math Soc 46, 389-408, 1940; Phil Trans Roy Soc Lond A238, 423-451, 1940. In this case Luke vol 1 section 5.11 looks as if it ought to do the job. But it's precisely the sort of tedious algebra I thought Maple was invented to free us from:-( John Harper, School of Mathematical and Computing Sciences, Victoria University, Wellington, New Zealand e-mail john.harper@vuw.ac.nz phone (+64)(4)463 5341 fax (+64)(4)463 5045 ============================================================================== From: kramsay@aol.commangled (Keith Ramsay) Subject: Re: Special Functions Date: 28 Dec 2000 18:59:25 GMT Newsgroups: sci.math Summary: [missing] In article <3A492101.5669663F@hot.rr.com>, "Ross A. Finlayson" writes: |I was thinking some more about the special functions. They have to do |with e and natural logarithms. I was wondering if there was a concise |definition of them. When I see references to "special functions", I think usually they don't mean exponential, log, trig, or inverse trig functions, but the ones "after" those, like the Gamma function, hypergeometric functions, elliptic functions, Airy functions, and so on. I don't know of a rigorous definition for the term "special functions", and I'm not sure there is one. They're solutions to differential equations that are considered natural objects of study. I've heard of people providing some kind of common theory to describe them in a more unified way. Painleve theory (if you have the right character set, that's "Painlevé") gives a kind of explanation for a bunch of them at least. Second order differential equations satisfying a "no movable essential singularities" condition were classified, and a bunch of the standard special functions appear in that classification. The more familiar e^x and log x fit into this scheme in a sense because they're solutions to linear ODE's not having moving essential signularities. The exponential function y=e^x solves y'=y, and it doesn't have any essential signularity on the complex plane. (On the Riemann sphere, it has one at infinity, but that one doesn't move as we change the initial conditions for y.) The natural log function y=log x solves xy'=1, and its essential singularity is at x=0 (a "branch point"). If we start it with a different initial condition at x=1, like y(1)=C, we get y=log x+C, whose essential signularity is still at x=0. Keith Ramsay