Series expansion around a singular point.


by muppet
Tags: expansion, point, series, singular
muppet
muppet is offline
#1
Dec7-12, 08:33 AM
P: 590
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
Going nuts? Turkey looks to pistachios to heat new eco-city
Space-tested fluid flow concept advances infectious disease diagnoses
SpaceX launches supplies to space station (Update)
mfb
mfb is offline
#2
Dec7-12, 09:52 AM
Mentor
P: 10,798
Use a Laurent series.
muppet
muppet is offline
#3
Dec7-12, 10:38 AM
P: 590
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
mfb is offline
#4
Dec7-12, 01:00 PM
Mentor
P: 10,798

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
muppet is offline
#5
Dec8-12, 10:34 AM
P: 590
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
bpet is offline
#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