About the Project

Watson expansions

AdvancedHelp

(0.001 seconds)

11—20 of 41 matching pages

11: 8.21 Generalized Sine and Cosine Integrals
§8.21(vi) Series Expansions
Power-Series Expansions
Spherical-Bessel-Function Expansions
For (8.21.16), (8.21.17), and further expansions in series of Bessel functions see Luke (1969b, pp. 56–57). …
§8.21(viii) Asymptotic Expansions
12: 22.11 Fourier and Hyperbolic Series
§22.11 Fourier and Hyperbolic Series
Similar expansions for cn 2 ( z , k ) and dn 2 ( z , k ) follow immediately from (22.6.1). … Again, similar expansions for cn 2 ( z , k ) and dn 2 ( z , k ) may be derived via (22.6.1). …
13: Bibliography S
  • A. Sidi (1985) Asymptotic expansions of Mellin transforms and analogues of Watson’s lemma. SIAM J. Math. Anal. 16 (4), pp. 896–906.
  • 14: 10.17 Asymptotic Expansions for Large Argument
    §10.17 Asymptotic Expansions for Large Argument
    §10.17(i) Hankel’s Expansions
    §10.17(ii) Asymptotic Expansions of Derivatives
    §10.17(v) Exponentially-Improved Expansions
    15: 10.19 Asymptotic Expansions for Large Order
    For higher coefficients in (10.19.8) in the case a = 0 (that is, in the expansions of J ν ( ν ) and Y ν ( ν ) ), see Watson (1944, §8.21), Temme (1997), and Jentschura and Lötstedt (2012). …
    16: 10.23 Sums
    For other types of expansions of arbitrary functions in series of Bessel functions, see Watson (1944, Chapters 17–19) and Erdélyi et al. (1953b, §§ 7.10.2–7.10.4). …
    17: 10.12 Generating Function and Associated Series
    Jacobi–Anger expansions: for z , θ , …
    18: 10.44 Sums
    For results analogous to (10.23.7) and (10.23.8) see Watson (1944, §§11.3 and 11.41).
    §10.44(iii) Neumann-Type Expansions
    19: 10.21 Zeros
    No two of the functions J 0 ( z ) , J 1 ( z ) , J 2 ( z ) , , have any common zeros other than z = 0 ; see Watson (1944, §15.28). …
    §10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros
    §10.21(vii) Asymptotic Expansions for Large Order
    The expansions (10.21.32)–(10.21.39) become progressively weaker as m increases. … See also Watson (1944, §§15.5, 15.51). …
    20: 12.9 Asymptotic Expansions for Large Variable
    §12.9 Asymptotic Expansions for Large Variable
    §12.9(i) Poincaré-Type Expansions
    12.9.1 U ( a , z ) e 1 4 z 2 z a 1 2 s = 0 ( 1 ) s ( 1 2 + a ) 2 s s ! ( 2 z 2 ) s , | ph z | 3 4 π δ ( < 3 4 π ) ,
    §12.9(ii) Bounds and Re-Expansions for the Remainder Terms
    Bounds and re-expansions for the error term in (12.9.1) can be obtained by use of (12.7.14) and §§13.7(ii), 13.7(iii). …