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Series expansion around a singular point. 
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#1
Dec712, 08:33 AM

P: 597

Hi All,
I have a problem involving some special functions (MeijerG 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. 


#2
Dec712, 09:52 AM

Mentor
P: 11,617

Use a Laurent series.



#3
Dec712, 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 


#4
Dec712, 01:00 PM

Mentor
P: 11,617

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 a_{3}=1 (not the only possibility), all other coefficients are 0. 


#5
Dec812, 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 Gfunction (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. 


#6
Dec912, 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 


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