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Why NO multiple Laurent series ?

  1. Aug 13, 2010 #1
    why NO multiple Laurent series ??

    why are ther Taylor series in several variables [tex] (x_{1} , x_{2} ,....., x_{n} [/tex] 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.
     
  2. jcsd
  3. Aug 15, 2010 #2

    HallsofIvy

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    Re: why NO multiple Laurent series ??

    ??? 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.
     
  4. Aug 15, 2010 #3
    Re: why NO multiple Laurent series ??

    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 [itex]z_{1}[/itex]), then the expansion coefficients are functions of the remaining arguments. Doing this successively, you will get the following symbolic expression:

    [tex]
    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}}}
    [/tex]

    where

    [tex]
    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}}}
    [/tex]
     
  5. Aug 15, 2010 #4
    Re: why NO multiple Laurent series ??

    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

    [tex] \iiint _{D}dxdydx Log(x+yzx^{4})artan(x+1+y+z) [/tex] then expanding into a multiple Laurent series in powers of x , y and z we can calculate [tex] \iiint dxdydz x^{m}y^{n}z^{k} [/tex] here 'D? is a rectangle on [tex] R^3 -(0)[/tex]
     
    Last edited: Aug 16, 2010
  6. Aug 23, 2010 #5
    Re: why NO multiple Laurent series ??

    and how about the CONVERGENCE ??

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