Series expansion around a singular point.

[muppet]

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. f(x)=x^{3/2}). 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:
f(x) =a + x^2 (b+ c Log[x])+ \ldots

where a,b, c are real numbers.

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

Thanks in advance.

 

[mfb]

Use a Laurent series.

 

[muppet]

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]

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]

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]

There's a bunch of series expansions listed at http://functions.wolfram.com/HypergeometricFunctions/MeijerG/06/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/HypergeometricFunctions/MeijerG/04/03/ and http://functions.wolfram.com/HypergeometricFunctions/MeijerG/04/04/

HTH