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Series expansion around a singular point.

by muppet
Tags: expansion, point, series, singular
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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.
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Dec7-12, 09:52 AM
P: 11,928
Use a Laurent series.
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

Dec7-12, 01:00 PM
P: 11,928
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.
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.
Dec9-12, 06:29 PM
P: 523
There's a bunch of series expansions listed at

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 (especially the discussion around algebraic and logarithmic branches) and and


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