Why NO multiple Laurent series ?

[zetafunction]

why NO multiple Laurent series ??

why are ther Taylor series in several variables $$(x_{1} , x_{2} ,..., x_{n}$$ but there are NO Laurent series in several variables ? why nobody has defined this series , or why they do not appear anywhere ?

i think there are PADE APPROXIMANTS in serveral variables but i have never NEVER heard of multiple Laurent series.

 

[HallsofIvy]

? There are. Of course, a complex function of a single complex variable already involves 4 real dimensions so functions of several complex variables are not normally covered in a first course in complex analysis.

 

[Dickfore]

A Taylor (Laurent) series in several variables ought to be thought of in the following way:

If you perform an expansion with respect to any of the arguments (say $z_{1}$), then the expansion coefficients are functions of the remaining arguments. Doing this successively, you will get the following symbolic expression:

$$f(z_{1}, \ldots, z_{n}) = \sum_{p_{1}, \ldots, p_{n} = -\infty}^{\infty}{K_{p_{1}, \ldots,p_{n}} \, (z_{1} - a_{1})^{p_{1}} \, \ldots \, (z_{n} - a_{n})^{p_{n}}}$$

where

$$K_{p_{1}, \ldots, p_{n}} = \frac{1}{(2 \pi i)^{n}} \, \oint_{C_{1}}{\ldots \oint_{C_{n}}{f(z_{1}, \ldots, z_{n}) \, (z_{1} - a_{1})^{-1-p_{1}} \, \ldots \, (z_{n} - a_{n})^{-1-p_{n}} \, dz_{1} \, \ldots \, dz_{n}}}$$

 

[zetafunction]

? There are. Of course, a complex function of a single complex variable already involves 4 real dimensions so functions of several complex variables are not normally covered in a first course in complex analysis.

oh , so you can have multi-variable LAURENT expansion ? , i thought there was some kind of mathematical restriction for them in the same way you can not define in general the inverse function in several variables ?

could you point me a book about an example of multi-variable Laurent series ? thanks a lot in advance

EDIT: i was thinking about this double Laurent series for the calculation of multiple integrals

$$\iiint _{D}dxdydx Log(x+yzx^{4})artan(x+1+y+z)$$ then expanding into a multiple Laurent series in powers of x , y and z we can calculate $$\iiint dxdydz x^{m}y^{n}z^{k}$$ here 'D? is a rectangle on $$R^3 -(0)$$

 

[zetafunction]

and how about the CONVERGENCE ??

given an analytic function $$f(z1,z2)$$ could you expand it into a CONVERGENT multiple Laurent series so it converges on the polydisc $$|z1| > 1$$ and $$|z2| >1$$ ??