How Can We Calculate Coefficients in Multivariable Laurent Series?

Click For Summary
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.

zetafunction
Messages
371
Reaction score
0
can we define a multivariable power series (laurent series)

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

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

how could i calculate the coefficients ?? a_{i,j,k,l} ?
 
Physics news on Phys.org
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.
 

Similar threads

Graduate Is the following sum a part of any known generalized function?
  • · Replies 2 ·
Replies
2
Views
2K
MHB How do you find the sum of an infinite series?
  • · Replies 7 ·
Replies
7
Views
2K
Graduate Solving a power series
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad How does the ratio test fail and the root test succeed here?
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad On convolution theorem of Laplace transform: Schiff
  • · Replies 4 ·
Replies
4
Views
3K
MHB How do I expand $\frac{e^{z^2}}{z^3}$ into a Laurent series?
  • · Replies 7 ·
Replies
7
Views
3K
High School Can We Simplify the Taylor Series Expansion for e^(f(x,y))?
  • · Replies 3 ·
Replies
3
Views
2K
MHB Calculate the integral using the Fourier coefficients
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Proving the Finite Binomial Series for k Non-Negative Integer
  • · Replies 9 ·
Replies
9
Views
2K
MHB Which Laurent Series for $\frac{1}{z(z+2)}$ in the Region $1 < |z-1| < 3$?
  • · Replies 26 ·
Replies
26
Views
5K