Register to reply

Series expansion around a singular point.

by muppet
Tags: expansion, point, series, singular
Share this thread:
muppet
#1
Dec7-12, 08:33 AM
P: 597
Hi All,

I have a problem involving some special functions (Meijer-G functions) that I'd like to approximate. At zero argument their first derivative vanishes, but their second and all higher derivatives vanish. (c.f. [itex]f(x)=x^{3/2}[/itex]). Playing about with some identities from Gradshteyn and Rhyzik, it looked to me as if this divergence goes like a negative fractional power of the argument, but I can ask Mathematica to give me a series expansion of the function about the origin, wherupon it returns something like:
[tex]f(x) =a + x^2 (b+ c Log[x])+ \ldots [/tex]

where a,b, c are real numbers.

How can I compute a "generalised taylor series" of this form analytically myself?

Thanks in advance.
Phys.Org News Partner Science news on Phys.org
Fungus deadly to AIDS patients found to grow on trees
Canola genome sequence reveals evolutionary 'love triangle'
Scientists uncover clues to role of magnetism in iron-based superconductors
mfb
#2
Dec7-12, 09:52 AM
Mentor
P: 11,831
Use a Laurent series.
muppet
#3
Dec7-12, 10:38 AM
P: 597
Thanks for your reply- wouldn't that only be useful for a pole of finite order? How could I use the Laurent series to extract the logarithmic coefficient?

Looking around a bit more it looks as if I want to compute something called the Puiseux series, which I'd never heard of before

mfb
#4
Dec7-12, 01:00 PM
Mentor
P: 11,831
Series expansion around a singular point.

Hmm, right, sqrt() has no analytic equivalent in the complex numbers, a Laurent series does not work.

Never heard of Puiseux series before, but your function is already one with n=2, and a3=1 (not the only possibility), all other coefficients are 0.
muppet
#5
Dec8-12, 10:34 AM
P: 597
To reiterate, what I wrote above is what mathematica gave me when I asked it to do a series expansion of a special function about the origin. The full function is a Meijer G-function (like a generalised hypergeometric function) and I'd like to know how to compute such expansions myself if at all possible.

This idea seems to be related to algebraic geometry somehow, so I might try another subforum. Thanks.
bpet
#6
Dec9-12, 06:29 PM
P: 523
There's a bunch of series expansions listed at http://functions.wolfram.com/Hyperge...6/ShowAll.html

I don't know how Mathematica calculated its series expansion but it might have applied a special case to one of the hypergeometric terms in the series.

Also possibly relevant is the type of branch singularity that occurs at 0; see http://en.wikipedia.org/wiki/Branch_point (especially the discussion around algebraic and logarithmic branches) and http://functions.wolfram.com/Hyperge...MeijerG/04/03/ and http://functions.wolfram.com/Hyperge...MeijerG/04/04/

HTH


Register to reply

Related Discussions
Series solution about a regular singular point (x=0) of xy''-xy'-y=0 Calculus & Beyond Homework 2
Series Solution around singular point Calculus & Beyond Homework 1
Convergence of a power series BEYOND a singular point? General Math 0
ODE Series Solution Near Regular Singular Point, x^2*y term? Calculus & Beyond Homework 3
Series solutions near a regular singular point Differential Equations 1