How Can We Calculate Coefficients in Multivariable Laurent Series?

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SUMMARY

The discussion focuses on calculating coefficients in multivariable Laurent series, specifically the expression ∑_{i,j,k,l,...=-∞}^{∞}a_{i,j,k,l,...}(X-a)^{i}(Y-b)^{j}(Z-c)^{k}(W-k)^{l}.... Participants explore the feasibility of defining such series with indices running over all integers, both positive and negative. The conversation emphasizes starting with simpler cases, such as nonnegative exponents or two-variable scenarios, before progressing to the full multivariable case.

PREREQUISITES
  • Understanding of multivariable calculus
  • Familiarity with Laurent series and power series
  • Knowledge of index notation in mathematical expressions
  • Basic skills in manipulating algebraic expressions
NEXT STEPS
  • Study the properties of Laurent series in detail
  • Learn how to calculate coefficients in simpler power series
  • Explore examples of multivariable power series with nonnegative exponents
  • Research techniques for handling infinite series and convergence
USEFUL FOR

Mathematicians, students of advanced calculus, and researchers working on series expansions in multiple variables will benefit from this discussion.

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can we define a multivariable power series (laurent series)

[tex]\sum_{i,j,k,l,...=-\infty}^{\infty}a_{i,j,k,l,...}(X-a)^{i}(Y-b)^{j}(Z-c)^{k}(W-k)^{l}...[/tex]

indices i,j k and l run over ALL the integers positive and negatives

how could i calculate the coefficients ?? [tex]a_{i,j,k,l}[/tex] ?
 
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Can you already do simpler cases? For example, only nonnegative exponents? Or just two variables? If not, work on those before you tackle this one.
 

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