About the Project
NIST

annulus

AdvancedHelp

(0.000 seconds)

3 matching pages

1: 2.10 Sums and Sequences
Let f ( z ) be analytic on the annulus 0 < | z | < r , with Laurent expansion
2.10.25 f ( z ) = n = - f n z n , 0 < | z | < r .
where 𝒞 is a simple closed contour in the annulus that encloses z = 0 . …
  • (c)

    The coefficients in the Laurent expansion

    2.10.27 g ( z ) = n = - g n z n , 0 < | z | < r ,

    have known asymptotic behavior as n ± .

  • 2.10.29 f n = g n + o ( r - n ) , n ± .
    2: 1.10 Functions of a Complex Variable
    §1.10(iii) Laurent Series
    Suppose f ( z ) is analytic in the annulus r 1 < | z - z 0 | < r 2 , 0 r 1 < r 2 , and r ( r 1 , r 2 ) . Then …The series (1.10.6) converges uniformly and absolutely on compact sets in the annulus. Let r 1 = 0 , so that the annulus becomes the punctured neighborhood N : 0 < | z - z 0 | < r 2 , and assume that f ( z ) is analytic in N , but not at z 0 . …
    3: 2.7 Differential Equations
    these series converging in an annulus | z | > a , with at least one of f 0 , g 0 , g 1 nonzero. …