Series expansion around a singular point.

In summary, Mathematica gave me a series expansion of a special function about the origin that includes the logarithmic coefficient.
  • #1
muppet
608
1
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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  • #3
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 :confused:
 
  • #4
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.
 
  • #5
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.
 
  • #6
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
 

1. What is a series expansion around a singular point?

A series expansion around a singular point is a mathematical technique used to approximate a function near a point where the function is not defined or is undefined. It involves representing the function as a sum of terms, each of which is a polynomial function of the variable, and then using the values of the function and its derivatives at the singular point to determine the coefficients of the terms.

2. What is the purpose of a series expansion around a singular point?

The purpose of a series expansion around a singular point is to extend the domain of a function and to approximate the behavior of the function near the singular point. This allows us to analyze and manipulate functions that are not defined or are undefined at certain points, and to gain insight into their behavior in these regions.

3. How do you determine the coefficients in a series expansion around a singular point?

The coefficients in a series expansion around a singular point are determined by using the values of the function and its derivatives at the singular point. These values are plugged into a formula that relates the coefficients to the derivatives. The more derivatives that are known, the more accurate the series expansion will be.

4. What is the relationship between series expansion and Taylor series expansion?

Series expansion around a singular point is a special case of Taylor series expansion. In series expansion around a singular point, the function is approximated using a polynomial series centered at the singular point, while in Taylor series expansion, the function is approximated using a polynomial series centered at a regular point. Series expansion around a singular point is used when the function is not defined or is undefined at the singular point, while Taylor series expansion is used when the function is defined and infinitely differentiable at the regular point.

5. How is series expansion around a singular point used in real-world applications?

Series expansion around a singular point is used in various fields of science and engineering, such as physics, chemistry, and biology, where functions may have undefined or discontinuous behavior at certain points. It is also used in numerical analysis and computer science to approximate functions and to solve differential equations. Additionally, series expansion around a singular point is used in financial modeling and risk management to approximate complex financial functions.

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