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Neumann-type expansions

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1: 10.44 Sums
§10.44(iii) Neumann-Type Expansions
2: 16.22 Asymptotic Expansions
§16.22 Asymptotic Expansions
Asymptotic expansions of G p , q m , n ( z ; 𝐚 ; 𝐛 ) for large z are given in Luke (1969a, §§5.7 and 5.10) and Luke (1975, §5.9). For asymptotic expansions of Meijer G -functions with large parameters see Fields (1973, 1983).
3: 20.8 Watson’s Expansions
§20.8 Watson’s Expansions
This reference and Bellman (1961, pp. 46–47) include other expansions of this type.
4: 4.47 Approximations
§4.47(i) Chebyshev-Series Expansions
5: 12.18 Methods of Computation
These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …
6: 29.16 Asymptotic Expansions
§29.16 Asymptotic Expansions
7: 6.20 Approximations
§6.20(ii) Expansions in Chebyshev Series
  • Luke and Wimp (1963) covers Ei ( x ) for x 4 (20D), and Si ( x ) and Ci ( x ) for x 4 (20D).

  • Luke (1969b, pp. 41–42) gives Chebyshev expansions of Ein ( a x ) , Si ( a x ) , and Cin ( a x ) for 1 x 1 , a . The coefficients are given in terms of series of Bessel functions.

  • Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric U -function (§13.2(i)) from which Chebyshev expansions near infinity for E 1 ( z ) , f ( z ) , and g ( z ) follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the U functions. If | ph z | < π the scheme can be used in backward direction.

  • §6.20(iii) Padé-Type and Rational Expansions
    8: 12.16 Mathematical Applications
    Integral transforms and sampling expansions are considered in Jerri (1982).
    9: 12.20 Approximations
    §12.20 Approximations
    Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions U ( a , b , x ) and M ( a , b , x ) 13.2(i)) whose regions of validity include intervals with endpoints x = and x = 0 , respectively. As special cases of these results a Chebyshev-series expansion for U ( a , x ) valid when λ x < follows from (12.7.14), and Chebyshev-series expansions for U ( a , x ) and V ( a , x ) valid when 0 x λ follow from (12.4.1), (12.4.2), (12.7.12), and (12.7.13). …
    10: 6.12 Asymptotic Expansions
    §6.12 Asymptotic Expansions
    §6.12(i) Exponential and Logarithmic Integrals
    For the function χ see §9.7(i). …
    §6.12(ii) Sine and Cosine Integrals