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1: 5.22 Tables
Abramowitz and Stegun (1964, Chapter 6) tabulates Γ ( x ) , ln Γ ( x ) , ψ ( x ) , and ψ ( x ) for x = 1 ( .005 ) 2 to 10D; ψ ′′ ( x ) and ψ ( 3 ) ( x ) for x = 1 ( .01 ) 2 to 10D; Γ ( n ) , 1 / Γ ( n ) , Γ ( n + 1 2 ) , ψ ( n ) , log 10 Γ ( n ) , log 10 Γ ( n + 1 3 ) , log 10 Γ ( n + 1 2 ) , and log 10 Γ ( n + 2 3 ) for n = 1 ( 1 ) 101 to 8–11S; Γ ( n + 1 ) for n = 100 ( 100 ) 1000 to 20S. …
2: 15.10 Hypergeometric Differential Equation
The ( 6 3 ) = 20 connection formulas for the principal branches of Kummer’s solutions are:
15.10.17 w 3 ( z ) = Γ ( 1 c ) Γ ( a + b c + 1 ) Γ ( a c + 1 ) Γ ( b c + 1 ) w 1 ( z ) + Γ ( c 1 ) Γ ( a + b c + 1 ) Γ ( a ) Γ ( b ) w 2 ( z ) ,
15.10.18 w 4 ( z ) = Γ ( 1 c ) Γ ( c a b + 1 ) Γ ( 1 a ) Γ ( 1 b ) w 1 ( z ) + Γ ( c 1 ) Γ ( c a b + 1 ) Γ ( c a ) Γ ( c b ) w 2 ( z ) ,
15.10.21 w 1 ( z ) = Γ ( c ) Γ ( c a b ) Γ ( c a ) Γ ( c b ) w 3 ( z ) + Γ ( c ) Γ ( a + b c ) Γ ( a ) Γ ( b ) w 4 ( z ) ,
15.10.25 w 1 ( z ) = Γ ( c ) Γ ( b a ) Γ ( b ) Γ ( c a ) w 5 ( z ) + Γ ( c ) Γ ( a b ) Γ ( a ) Γ ( c b ) w 6 ( z ) ,
3: 5.11 Asymptotic Expansions
The scaled gamma function Γ ( z ) is defined in (5.11.3) and its main property is Γ ( z ) 1 as z in the sector | ph z | π δ . Wrench (1968) gives exact values of g k up to g 20 . … In this subsection a , b , and c are real or complex constants. …
5.11.12 Γ ( z + a ) Γ ( z + b ) z a b ,
5.11.19 Γ ( z + a ) Γ ( z + b ) Γ ( z + c ) k = 0 ( 1 ) k ( c a ) k ( c b ) k k ! Γ ( a + b c + z k ) .
4: 8.26 Tables
  • Khamis (1965) tabulates P ( a , x ) for a = 0.05 ( .05 ) 10 ( .1 ) 20 ( .25 ) 70 , 0.0001 x 250 to 10D.

  • Zhang and Jin (1996, Table 3.8) tabulates γ ( a , x ) for a = 0.5 , 1 , 3 , 5 , 10 , 25 , 50 , 100 , x = 0 ( .1 ) 1 ( 1 ) 3 , 5 ( 5 ) 30 , 50 , 100 to 8D or 8S.

  • Abramowitz and Stegun (1964, pp. 245–248) tabulates E n ( x ) for n = 2 , 3 , 4 , 10 , 20 , x = 0 ( .01 ) 2 to 7D; also ( x + n ) e x E n ( x ) for n = 2 , 3 , 4 , 10 , 20 , x 1 = 0 ( .01 ) 0.1 ( .05 ) 0.5 to 6S.

  • Pagurova (1961) tabulates E n ( x ) for n = 0 ( 1 ) 20 , x = 0 ( .01 ) 2 ( .1 ) 10 to 4-9S; e x E n ( x ) for n = 2 ( 1 ) 10 , x = 10 ( .1 ) 20 to 7D; e x E p ( x ) for p = 0 ( .1 ) 1 , x = 0.01 ( .01 ) 7 ( .05 ) 12 ( .1 ) 20 to 7S or 7D.

  • Zhang and Jin (1996, Table 19.1) tabulates E n ( x ) for n = 1 , 2 , 3 , 5 , 10 , 15 , 20 , x = 0 ( .1 ) 1 , 1.5 , 2 , 3 , 5 , 10 , 20 , 30 , 50 , 100 to 7D or 8S.

  • 5: 20.10 Integrals
    20.10.1 0 x s 1 θ 2 ( 0 | i x 2 ) d x = 2 s ( 1 2 s ) π s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 1 ,
    20.10.2 0 x s 1 ( θ 3 ( 0 | i x 2 ) 1 ) d x = π s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 1 ,
    20.10.3 0 x s 1 ( 1 θ 4 ( 0 | i x 2 ) ) d x = ( 1 2 1 s ) π s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 0 .
    Let s , , and β be constants such that s > 0 , > 0 , and | β | + | β | . …
    6: 25.5 Integral Representations
    25.5.1 ζ ( s ) = 1 Γ ( s ) 0 x s 1 e x 1 d x , s > 1 .
    25.5.13 ζ ( s ) = π s / 2 s ( s 1 ) Γ ( 1 2 s ) + π s / 2 Γ ( 1 2 s ) 1 ( x s / 2 + x ( 1 s ) / 2 ) ω ( x ) x d x , s 1 ,
    In (25.5.15)–(25.5.19), 0 < s < 1 , ψ ( x ) is the digamma function, and γ is Euler’s constant5.2). …
    25.5.17 ζ ( 1 + s ) = sin ( π s ) π 0 ( γ + ψ ( 1 + x ) ) x s 1 d x ,
    25.5.19 ζ ( m + s ) = ( 1 ) m 1 Γ ( s ) sin ( π s ) π Γ ( m + s ) 0 ψ ( m ) ( 1 + x ) x s d x , m = 1 , 2 , 3 , .
    7: Bibliography
  • M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.
  • A. Abramov (1960) Tables of ln Γ ( z ) for Complex Argument. Pergamon Press, New York.
  • A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.
  • D. E. Amos (1989) Repeated integrals and derivatives of K Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.
  • H. Appel (1968) Numerical Tables for Angular Correlation Computations in α -, β - and γ -Spectroscopy: 3 j -, 6 j -, 9 j -Symbols, F- and Γ -Coefficients. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology, Springer-Verlag.
  • 8: 25.20 Approximations
  • Cody et al. (1971) gives rational approximations for ζ ( s ) in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are 0.5 s 5 , 5 s 11 , 11 s 25 , 25 s 55 . Precision is varied, with a maximum of 20S.

  • Antia (1993) gives minimax rational approximations for Γ ( s + 1 ) F s ( x ) , where F s ( x ) is the Fermi–Dirac integral (25.12.14), for the intervals < x 2 and 2 x < , with s = 1 2 , 1 2 , 3 2 , 5 2 . For each s there are three sets of approximations, with relative maximum errors 10 4 , 10 8 , 10 12 .

  • 9: 30.9 Asymptotic Approximations and Expansions
    §30.9(i) Prolate Spheroidal Wave Functions
    As γ 2 + , with q = 2 ( n m ) + 1 , … The asymptotic behavior of λ n m ( γ 2 ) and a n , k m ( γ 2 ) as n in descending powers of 2 n + 1 is derived in Meixner (1944). …The asymptotic behavior of 𝖯𝗌 n m ( x , γ 2 ) and 𝖰𝗌 n m ( x , γ 2 ) as x ± 1 is given in Erdélyi et al. (1955, p. 151). The behavior of λ n m ( γ 2 ) for complex γ 2 and large | λ n m ( γ 2 ) | is investigated in Hunter and Guerrieri (1982). …
    10: 11.6 Asymptotic Expansions
    11.6.1 𝐊 ν ( z ) 1 π k = 0 Γ ( k + 1 2 ) ( 1 2 z ) ν 2 k 1 Γ ( ν + 1 2 k ) , | ph z | π δ ,
    where δ is an arbitrary small positive constant. …
    11.6.2 𝐌 ν ( z ) 1 π k = 0 ( 1 ) k + 1 Γ ( k + 1 2 ) ( 1 2 z ) ν 2 k 1 Γ ( ν + 1 2 k ) , | ph z | 1 2 π δ .
    where γ is Euler’s constant5.2(ii)). …
    c 3 ( λ ) = 20 λ 6 4 λ 4 ,