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11: 30.7 Graphics
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Figure 30.7.7: 𝖯𝗌 n 1 ( x , 30 ) , n = 1 , 2 , 3 , 4 , 1 x 1 . Magnify
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Figure 30.7.8: 𝖯𝗌 n 1 ( x , 30 ) , n = 1 , 2 , 3 , 4 , 1 x 1 . Magnify
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Figure 30.7.10: 𝖯𝗌 3 1 ( x , γ 2 ) , 1 x 1 , 50 γ 2 50 . Magnify 3D Help
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Figure 30.7.13: 𝖰𝗌 n 1 ( x , 4 ) , for n = 1 , 2 , 3 , 4 , 1 < x < 1 . Magnify
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Figure 30.7.14: 𝖰𝗌 n 1 ( x , 4 ) , n = 1 , 2 , 3 , 4 , 1 < x < 1 . Magnify
12: 12.19 Tables
  • Abramowitz and Stegun (1964, Chapter 19) includes U ( a , x ) and V ( a , x ) for ± a = 0 ( .1 ) 1 ( .5 ) 5 , x = 0 ( .1 ) 5 , 5S; W ( a , ± x ) for ± a = 0 ( .1 ) 1 ( 1 ) 5 , x = 0 ( .1 ) 5 , 4-5D or 4-5S.

  • Miller (1955) includes W ( a , x ) , W ( a , x ) , and reduced derivatives for a = 10 ( 1 ) 10 , x = 0 ( .1 ) 10 , 8D or 8S. Modulus and phase functions, and also other auxiliary functions are tabulated.

  • Fox (1960) includes modulus and phase functions for W ( a , x ) and W ( a , x ) , and several auxiliary functions for x 1 = 0 ( .005 ) 0.1 , a = 10 ( 1 ) 10 , 8S.

  • Kireyeva and Karpov (1961) includes D p ( x ( 1 + i ) ) for ± x = 0 ( .1 ) 5 , p = 0 ( .1 ) 2 , and ± x = 5 ( .01 ) 10 , p = 0 ( .5 ) 2 , 7D.

  • Murzewski and Sowa (1972) includes D n ( x ) ( = U ( n 1 2 , x ) ) for n = 1 ( 1 ) 20 , x = 0 ( .05 ) 3 , 7S.

  • 13: 6.19 Tables
  • Abramowitz and Stegun (1964, Chapter 5) includes x 1 Si ( x ) , x 2 Cin ( x ) , x 1 Ein ( x ) , x 1 Ein ( x ) , x = 0 ( .01 ) 0.5 ; Si ( x ) , Ci ( x ) , Ei ( x ) , E 1 ( x ) , x = 0.5 ( .01 ) 2 ; Si ( x ) , Ci ( x ) , x e x Ei ( x ) , x e x E 1 ( x ) , x = 2 ( .1 ) 10 ; x f ( x ) , x 2 g ( x ) , x e x Ei ( x ) , x e x E 1 ( x ) , x 1 = 0 ( .005 ) 0.1 ; Si ( π x ) , Cin ( π x ) , x = 0 ( .1 ) 10 . Accuracy varies but is within the range 8S–11S.

  • Zhang and Jin (1996, pp. 652, 689) includes Si ( x ) , Ci ( x ) , x = 0 ( .5 ) 20 ( 2 ) 30 , 8D; Ei ( x ) , E 1 ( x ) , x = [ 0 , 100 ] , 8S.

  • Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of z e z E 1 ( z ) , x = 19 ( 1 ) 20 , y = 0 ( 1 ) 20 , 6D; e z E 1 ( z ) , x = 4 ( .5 ) 2 , y = 0 ( .2 ) 1 , 6D; E 1 ( z ) + ln z , x = 2 ( .5 ) 2.5 , y = 0 ( .2 ) 1 , 6D.

  • Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of E 1 ( z ) , ± x = 0.5 , 1 , 3 , 5 , 10 , 15 , 20 , 50 , 100 , y = 0 ( .5 ) 1 ( 1 ) 5 ( 5 ) 30 , 50 , 100 , 8S.

  • 14: 14.33 Tables
  • Abramowitz and Stegun (1964, Chapter 8) tabulates 𝖯 n ( x ) for n = 0 ( 1 ) 3 , 9 , 10 , x = 0 ( .01 ) 1 , 5–8D; 𝖯 n ( x ) for n = 1 ( 1 ) 4 , 9 , 10 , x = 0 ( .01 ) 1 , 5–7D; 𝖰 n ( x ) and 𝖰 n ( x ) for n = 0 ( 1 ) 3 , 9 , 10 , x = 0 ( .01 ) 1 , 6–8D; P n ( x ) and P n ( x ) for n = 0 ( 1 ) 5 , 9 , 10 , x = 1 ( .2 ) 10 , 6S; Q n ( x ) and Q n ( x ) for n = 0 ( 1 ) 3 , 9 , 10 , x = 1 ( .2 ) 10 , 6S. (Here primes denote derivatives with respect to x .)

  • Zhang and Jin (1996, Chapter 4) tabulates 𝖯 n ( x ) for n = 2 ( 1 ) 5 , 10 , x = 0 ( .1 ) 1 , 7D; 𝖯 n ( cos θ ) for n = 1 ( 1 ) 4 , 10 , θ = 0 ( 5 ) 90 , 8D; 𝖰 n ( x ) for n = 0 ( 1 ) 2 , 10 , x = 0 ( .1 ) 0.9 , 8S; 𝖰 n ( cos θ ) for n = 0 ( 1 ) 3 , 10 , θ = 0 ( 5 ) 90 , 8D; 𝖯 n m ( x ) for m = 1 ( 1 ) 4 , n m = 0 ( 1 ) 2 , n = 10 , x = 0 , 0.5 , 8S; 𝖰 n m ( x ) for m = 1 ( 1 ) 4 , n = 0 ( 1 ) 2 , 10 , 8S; 𝖯 ν m ( cos θ ) for m = 0 ( 1 ) 3 , ν = 0 ( .25 ) 5 , θ = 0 ( 15 ) 90 , 5D; P n ( x ) for n = 2 ( 1 ) 5 , 10 , x = 1 ( 1 ) 10 , 7S; Q n ( x ) for n = 0 ( 1 ) 2 , 10 , x = 2 ( 1 ) 10 , 8S. Corresponding values of the derivative of each function are also included, as are 6D values of the first 5 ν -zeros of 𝖯 ν m ( cos θ ) and of its derivative for m = 0 ( 1 ) 4 , θ = 10 , 30 , 150 .

  • Žurina and Karmazina (1964, 1965) tabulate the conical functions 𝖯 1 2 + i τ ( x ) for τ = 0 ( .01 ) 50 , x = 0.9 ( .1 ) 0.9 , 7S; P 1 2 + i τ ( x ) for τ = 0 ( .01 ) 50 , x = 1.1 ( .1 ) 2 ( .2 ) 5 ( .5 ) 10 ( 10 ) 60 , 7D. Auxiliary tables are included to facilitate computation for larger values of τ when 1 < x < 1 .

  • Žurina and Karmazina (1963) tabulates the conical functions 𝖯 1 2 + i τ 1 ( x ) for τ = 0 ( .01 ) 25 , x = 0.9 ( .1 ) 0.9 , 7S; P 1 2 + i τ 1 ( x ) for τ = 0 ( .01 ) 25 , x = 1.1 ( .1 ) 2 ( .2 ) 5 ( .5 ) 10 ( 10 ) 60 , 7S. Auxiliary tables are included to assist computation for larger values of τ when 1 < x < 1 .

  • 15: 18.6 Symmetry, Special Values, and Limits to Monomials
    Table 18.6.1: Classical OP’s: symmetry and special values.
    p n ( x ) p n ( x ) p n ( 1 ) p 2 n ( 0 ) p 2 n + 1 ( 0 )
    T n ( x ) ( 1 ) n T n ( x ) 1 ( 1 ) n ( 1 ) n ( 2 n + 1 )
    V n ( x ) ( 1 ) n W n ( x ) 1 ( 1 ) n ( 1 ) n ( 2 n + 2 )
    P n ( x ) ( 1 ) n P n ( x ) 1 ( 1 ) n ( 1 2 ) n / n ! 2 ( 1 ) n ( 1 2 ) n + 1 / n !
    18.6.2 lim α P n ( α , β ) ( x ) P n ( α , β ) ( 1 ) = ( 1 + x 2 ) n ,
    18.6.5 lim α L n ( α ) ( α x ) L n ( α ) ( 0 ) = ( 1 x ) n .
    16: 18.5 Explicit Representations
    Table 18.5.1: Classical OP’s: Rodrigues formulas (18.5.5).
    p n ( x ) F ( x ) κ n
    P n ( x ) 1 x 2 ( 2 ) n n !
    H n ( x ) 1 ( 1 ) n
    𝐻𝑒 n ( x ) 1 ( 1 ) n
    The first of each of equations (18.5.7) and (18.5.8) can be regarded as definitions of P n ( α , β ) ( x ) when the conditions α > 1 and β > 1 are not satisfied. …
    L 1 ( x ) = x + 1 ,
    17: 14.3 Definitions and Hypergeometric Representations
    §14.3(i) Interval 1 < x < 1
    The following are real-valued solutions of (14.2.2) when μ , ν and x ( 1 , 1 ) . …
    §14.3(ii) Interval 1 < x <
    The following are solutions of (14.2.2) when μ , ν and x > 1 . … Like P ν μ ( x ) , but unlike Q ν μ ( x ) , 𝑸 ν μ ( x ) is real-valued when ν , μ and x ( 1 , ) , and is defined for all values of ν and μ . …
    18: 5.23 Approximations
    Hart et al. (1968) gives minimax polynomial and rational approximations to Γ ( x ) and ln Γ ( x ) in the intervals 0 x 1 , 8 x 1000 , 12 x 1000 ; precision is variable. … Luke (1969b) gives the coefficients to 20D for the Chebyshev-series expansions of Γ ( 1 + x ) , 1 / Γ ( 1 + x ) , Γ ( x + 3 ) , ln Γ ( x + 3 ) , ψ ( x + 3 ) , and the first six derivatives of ψ ( x + 3 ) for 0 x 1 . …Clenshaw (1962) also gives 20D Chebyshev-series coefficients for Γ ( 1 + x ) and its reciprocal for 0 x 1 . …
    19: 16.14 Partial Differential Equations
    x ( 1 x ) 2 F 1 x 2 + y ( 1 x ) 2 F 1 x y + ( γ ( α + β + 1 ) x ) F 1 x β y F 1 y α β F 1 = 0 ,
    x ( 1 x ) 2 F 2 x 2 x y 2 F 2 x y + ( γ ( α + β + 1 ) x ) F 2 x β y F 2 y α β F 2 = 0 ,
    x ( 1 x ) 2 F 3 x 2 + y 2 F 3 x y + ( γ ( α + β + 1 ) x ) F 3 x α β F 3 = 0 ,
    x ( 1 x ) 2 F 4 x 2 2 x y 2 F 4 x y y 2 2 F 4 y 2 + ( γ ( α + β + 1 ) x ) F 4 x ( α + β + 1 ) y F 4 y α β F 4 = 0 ,
    (The region of convergence | x | + | y | < 1 4 is not quite maximal.) …
    20: 10.75 Tables
  • The main tables in Abramowitz and Stegun (1964, Chapter 9) give J 0 ( x ) to 15D, J 1 ( x ) , J 2 ( x ) , Y 0 ( x ) , Y 1 ( x ) to 10D, Y 2 ( x ) to 8D, x = 0 ( .1 ) 17.5 ; Y n ( x ) ( 2 / π ) J n ( x ) ln x , n = 0 , 1 , x = 0 ( .1 ) 2 , 8D; J n ( x ) , Y n ( x ) , n = 3 ( 1 ) 9 , x = 0 ( .2 ) 20 , 5D or 5S; J n ( x ) , Y n ( x ) , n = 0 ( 1 ) 20 ( 10 ) 50 , 100 , x = 1 , 2 , 5 , 10 , 50 , 100 , 10S; modulus and phase functions x M n ( x ) , θ n ( x ) x , n = 0 , 1 , 2 , 1 / x = 0 ( .01 ) 0.1 , 8D.

  • Achenbach (1986) tabulates J 0 ( x ) , J 1 ( x ) , Y 0 ( x ) , Y 1 ( x ) , x = 0 ( .1 ) 8 , 20D or 18–20S.

  • Achenbach (1986) tabulates I 0 ( x ) , I 1 ( x ) , K 0 ( x ) , K 1 ( x ) , x = 0 ( .1 ) 8 , 19D or 19–21S.

  • Bickley and Nayler (1935) tabulates Ki n ( x ) 10.43(iii)) for n = 1 ( 1 ) 16 , x = 0 ( .05 ) 0.2 ( .1 ) 2, 3, 9D.

  • The main tables in Abramowitz and Stegun (1964, Chapter 10) give 𝗃 n ( x ) , 𝗒 n ( x ) n = 0 ( 1 ) 8 , x = 0 ( .1 ) 10 , 5–8S; 𝗃 n ( x ) , 𝗒 n ( x ) n = 0 ( 1 ) 20 ( 10 ) 50 , 100, x = 1 , 2 , 5 , 10 , 50 , 100 , 10S; 𝗂 n ( 1 ) ( x ) , 𝗄 n ( x ) , n = 0 , 1 , 2 , x = 0 ( .1 ) 5 , 4–9D; 𝗂 n ( 1 ) ( x ) , 𝗄 n ( x ) , n = 0 ( 1 ) 20 ( 10 ) 50 , 100, x = 1 , 2 , 5 , 10 , 50 , 100 , 10S. (For the notation see §10.1 and §10.47(ii).)