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1: 16.11 Asymptotic Expansions
§16.11(i) Formal Series
For subsequent use we define two formal infinite series, E p , q ( z ) and H p , q ( z ) , as follows:
16.11.1 E p , q ( z ) = ( 2 π ) ( p - q ) / 2 κ - ν - ( 1 / 2 ) e κ z 1 / κ k = 0 c k ( κ z 1 / κ ) ν - k , p < q + 1 ,
16.11.2 H p , q ( z ) = m = 1 p k = 0 ( - 1 ) k k ! Γ ( a m + k ) ( = 1 m p Γ ( a - a m - k ) / = 1 q Γ ( b - a m - k ) ) z - a m - k .
The formal series (16.11.2) for H q + 1 , q ( z ) converges if | z | > 1 , and …
2: 10.70 Zeros
Let μ = 4 ν 2 and f ( t ) denote the formal series
3: 3.10 Continued Fractions
Every convergent, asymptotic, or formal seriesWe say that it corresponds to the formal power seriesWe say that it is associated with the formal power series f ( z ) in (3.10.7) if the expansion of its n th convergent C n in ascending powers of z , agrees with (3.10.7) up to and including the term in z 2 n - 1 , n = 1 , 2 , 3 , . …
4: 1.17 Integral and Series Representations of the Dirac Delta
By analogy with §1.17(ii) we have the formal series representation …
5: 1.8 Fourier Series
§1.8 Fourier Series
Formally, … …
Uniqueness of Fourier Series
6: 16.5 Integral Representations and Integrals
In this event, the formal power-series expansion of the left-hand side (obtained from (16.2.1)) is the asymptotic expansion of the right-hand side as z 0 in the sector | ph ( - z ) | ( p + 1 - q - δ ) π / 2 , where δ is an arbitrary small positive constant. …
7: 2.1 Definitions and Elementary Properties
Let a s x - s be a formal power series (convergent or divergent) and for each positive integer n , …
8: Bibliography B
  • L. J. Billera, C. Greene, R. Simion, and R. P. Stanley (Eds.) (1996) Formal Power Series and Algebraic Combinatorics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 24, American Mathematical Society, Providence, RI.
  • 9: 16.4 Argument Unity
    Contiguous balanced series have parameters shifted by an integer but still balanced. … … when the series on the right terminates and the series on the left converges. …
    §16.4(v) Bilateral Series
    Denote, formally, the bilateral hypergeometric function …
    10: 3.11 Approximation Techniques
    be a formal power series. …