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1: 13.30 Tables
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  • Žurina and Osipova (1964) tabulates M ⁡ ( a , b , x ) and U ⁡ ( a , b , x ) for b = 2 , a = 0.98 ⁢ ( .02 ) ⁢ 1.10 , x = 0 ⁢ ( .01 ) ⁢ 4 , 7D or 7S.

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  • Slater (1960) tabulates M ⁡ ( a , b , x ) for a = 1 ⁢ ( .1 ) ⁢ 1 , b = 0.1 ⁢ ( .1 ) ⁢ 1 , and x = 0.1 ⁢ ( .1 ) ⁢ 10 , 7–9S; M ⁡ ( a , b , 1 ) for a = 11 ⁢ ( .2 ) ⁢ 2 and b = 4 ⁢ ( .2 ) ⁢ 1 , 7D; the smallest positive x -zero of M ⁡ ( a , b , x ) for a = 4 ⁢ ( .1 ) 0.1 and b = 0.1 ⁢ ( .1 ) ⁢ 2.5 , 7D.

  • 2: 14.33 Tables
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  • 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 .)

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  • 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 ∘ .

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  • 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 .

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  • 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 .

  • 3: 33.24 Tables
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  • Curtis (1964a) tabulates P ℓ ⁡ ( ϵ , r ) , Q ℓ ⁡ ( ϵ , r ) 33.1), and related functions for ℓ = 0 , 1 , 2 and ϵ = 2 ⁢ ( .2 ) ⁢ 2 , with x = 0 ⁢ ( .1 ) ⁢ 4 for ϵ < 0 and x = 0 ⁢ ( .1 ) ⁢ 10 for ϵ 0 ; 6D.

  • 4: Bibliography M
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  • A. J. MacLeod (1998) Algorithm 779: Fermi-Dirac functions of order 1 / 2 , 1 / 2 , 3 / 2 , 5 / 2 . ACM Trans. Math. Software 24 (1), pp. 1–12.
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  • A. R. Miller (2003) On a Kummer-type transformation for the generalized hypergeometric function F 2 2 . J. Comput. Appl. Math. 157 (2), pp. 507–509.
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  • R. Milson (2017) Exceptional orthogonal polynomials.
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  • P. M. Morse and H. Feshbach (1953a) Methods of Theoretical Physics. Vol. 1, McGraw-Hill Book Co., New York.
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  • P. M. Morse and H. Feshbach (1953b) Methods of Theoretical Physics. Vol. 2, McGraw-Hill Book Co., New York.
  • 5: 6.19 Tables
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  • 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.

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  • 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.

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  • 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.

  • 6: 19.37 Tables
    ►Tabulated for k 2 = 0 ⁢ ( .001 ) ⁢ 1 to 8D by BeliÍ¡akov et al. (1962). … ►
    Functions R F ⁡ ( x 2 , 1 , y 2 ) and R G ⁡ ( x 2 , 1 , y 2 )
    ►Tabulated for x = 0 ⁢ ( .1 ) ⁢ 1 , y = 1 ⁢ ( .2 ) ⁢ 6 to 3D by Nellis and Carlson (1966). ►
    Function R F ⁡ ( a 2 , b 2 , c 2 ) with a ⁢ b ⁢ c = 1
    ►Tabulated for σ = 0 ⁢ ( .05 ) ⁢ 0.5 ⁢ ( .1 ) ⁢ 1 ⁢ ( .2 ) ⁢ 2 ⁢ ( .5 ) ⁢ 5 , cos ⁡ ( 3 ⁢ γ ) = 1 ⁢ ( .2 ) ⁢ 1 to 5D by Carlson (1961a). …
    7: 11.14 Tables
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  • Abramowitz and Stegun (1964, Chapter 12) tabulates 0 x ( I 0 ⁡ ( t ) 𝐋 0 ⁡ ( t ) ) ⁢ d t and ( 2 / π ) ⁢ x t 1 ⁢ 𝐇 0 ⁡ ( t ) ⁢ d t for x = 0 ⁢ ( .1 ) ⁢ 5 to 5D or 7D; 0 x ( 𝐇 0 ⁡ ( t ) Y 0 ⁡ ( t ) ) ⁢ d t ( 2 / π ) ⁢ ln ⁡ x , 0 x ( I 0 ⁡ ( t ) 𝐋 0 ⁡ ( t ) ) ⁢ d t ( 2 / π ) ⁢ ln ⁡ x , and x t 1 ⁢ ( 𝐇 0 ⁡ ( t ) Y 0 ⁡ ( t ) ) ⁢ d t for x 1 = 0 ⁢ ( .01 ) ⁢ 0.2 to 6D.

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  • Jahnke and Emde (1945) tabulates 𝐄 n ⁡ ( x ) for n = 1 , 2 and x = 0 ⁢ ( .01 ) ⁢ 14.99 to 4D.

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  • Agrest and Maksimov (1971, Chapter 11) defines incomplete Struve, Anger, and Weber functions and includes tables of an incomplete Struve function 𝐇 n ⁡ ( x , α ) for n = 0 , 1 , x = 0 ⁢ ( .2 ) ⁢ 10 , and α = 0 ⁢ ( .2 ) ⁢ 1.4 , 1 2 ⁢ π , together with surface plots.

  • 8: 28.35 Tables
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  • Blanch and Rhodes (1955) includes 𝐵𝑒 n ⁡ ( t ) , 𝐵𝑜 n ⁡ ( t ) , t = 1 2 ⁢ q , n = 0 ⁢ ( 1 ) ⁢ 15 ; 8D. The range of t is 0 to 0.1, with step sizes ranging from 0.002 down to 0.00025. Notation: 𝐵𝑒 n ⁡ ( t ) = a n ⁡ ( q ) + 2 ⁢ q ( 4 ⁢ n + 2 ) ⁢ q , 𝐵𝑜 n ⁡ ( t ) = b n ⁡ ( q ) + 2 ⁢ q ( 4 ⁢ n 2 ) ⁢ q .

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  • National Bureau of Standards (1967) includes the eigenvalues a n ⁡ ( q ) , b n ⁡ ( q ) for n = 0 ⁢ ( 1 ) ⁢ 3 with q = 0 ⁢ ( .2 ) ⁢ 20 ⁢ ( .5 ) ⁢ 37 ⁢ ( 1 ) ⁢ 100 , and n = 4 ⁢ ( 1 ) ⁢ 15 with q = 0 ⁢ ( 2 ) ⁢ 100 ; Fourier coefficients for ce n ⁡ ( x , q ) and se n ⁡ ( x , q ) for n = 0 ⁢ ( 1 ) ⁢ 15 , n = 1 ⁢ ( 1 ) ⁢ 15 , respectively, and various values of q in the interval [ 0 , 100 ] ; joining factors g e , n ⁡ ( q ) , f e , n ⁡ ( q ) for n = 0 ⁢ ( 1 ) ⁢ 15 with q = 0 ⁢ ( .5 ⁢  to  ⁢ 10 ) ⁢ 100 (but in a different notation). Also, eigenvalues for large values of q . Precision is generally 8D.

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  • Stratton et al. (1941) includes b n , b n , and the corresponding Fourier coefficients for Se n ⁡ ( c , x ) and So n ⁡ ( c , x ) for n = 0 or 1 ⁢ ( 1 ) ⁢ 4 , c = 0 ⁢ ( .1 ⁢ or ⁢ .2 ) ⁢ 4.5 . Precision is mostly 5S. Notation: c = 2 ⁢ q , b n = a n + 2 ⁢ q , b n = b n + 2 ⁢ q , and for Se n ⁡ ( c , x ) , So n ⁡ ( c , x ) see §28.1.

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  • Zhang and Jin (1996, pp. 521–532) includes the eigenvalues a n ⁡ ( q ) , b n + 1 ⁡ ( q ) for n = 0 ⁢ ( 1 ) ⁢ 4 , q = 0 ⁢ ( 1 ) ⁢ 50 ; n = 0 ⁢ ( 1 ) ⁢ 20 ( a ’s) or 19 ( b ’s), q = 1 , 3 , 5 , 10 , 15 , 25 , 50 ⁢ ( 50 ) ⁢ 200 . Fourier coefficients for ce n ⁡ ( x , 10 ) , se n + 1 ⁡ ( x , 10 ) , n = 0 ⁢ ( 1 ) ⁢ 7 . Mathieu functions ce n ⁡ ( x , 10 ) , se n + 1 ⁡ ( x , 10 ) , and their first x -derivatives for n = 0 ⁢ ( 1 ) ⁢ 4 , x = 0 ⁢ ( 5 ∘ ) ⁢ 90 ∘ . Modified Mathieu functions Mc n ( j ) ⁡ ( x , 10 ) , Ms n + 1 ( j ) ⁡ ( x , 10 ) , and their first x -derivatives for n = 0 ⁢ ( 1 ) ⁢ 4 , j = 1 , 2 , x = 0 ⁢ ( .2 ) ⁢ 4 . Precision is mostly 9S.

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  • Ince (1932) includes the first zero for ce n , se n for n = 2 ⁢ ( 1 ) ⁢ 5 or 6 , q = 0 ⁢ ( 1 ) ⁢ 10 ⁢ ( 2 ) ⁢ 40 ; 4D. This reference also gives zeros of the first derivatives, together with expansions for small q .

  • 9: 8.26 Tables
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  • Pearson (1965) tabulates the function I ⁡ ( u , p ) ( = P ⁡ ( p + 1 , u ) ) for p = 1 ⁢ ( .05 ) ⁢ 0 ⁢ ( .1 ) ⁢ 5 ⁢ ( .2 ) ⁢ 50 , u = 0 ⁢ ( .1 ) ⁢ u p to 7D, where I ⁡ ( u , u p ) rounds off to 1 to 7D; also I ⁡ ( u , p ) for p = 0.75 ⁢ ( .01 ) 1 , u = 0 ⁢ ( .1 ) ⁢ 6 to 5D.

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  • 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.

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  • Chiccoli et al. (1988) presents a short table of E p ⁡ ( x ) for p = 9 2 ⁢ ( 1 ) 1 2 , 0 x 200 to 14S.

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  • 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.

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  • 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.

  • 10: 7.23 Tables
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  • Abramowitz and Stegun (1964, Chapter 7) includes erf ⁡ x , ( 2 / π ) ⁢ e x 2 , x [ 0 , 2 ] , 10D; ( 2 / π ) ⁢ e x 2 , x [ 2 , 10 ] , 8S; x ⁢ e x 2 ⁢ erfc ⁡ x , x 2 [ 0 , 0.25 ] , 7D; 2 n ⁢ Γ ⁡ ( 1 2 ⁢ n + 1 ) ⁢ i n ⁢ erfc ⁡ ( x ) , n = 1 ⁢ ( 1 ) ⁢ 6 , 10 , 11 , x [ 0 , 5 ] , 6S; F ⁡ ( x ) , x [ 0 , 2 ] , 10D; x ⁢ F ⁡ ( x ) , x 2 [ 0 , 0.25 ] , 9D; C ⁡ ( x ) , S ⁡ ( x ) , x [ 0 , 5 ] , 7D; f ⁡ ( x ) , g ⁡ ( x ) , x [ 0 , 1 ] , x 1 [ 0 , 1 ] , 15D.

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  • Zhang and Jin (1996, pp. 637, 639) includes ( 2 / π ) ⁢ e x 2 , erf ⁡ x , x = 0 ⁢ ( .02 ) ⁢ 1 ⁢ ( .04 ) ⁢ 3 , 8D; C ⁡ ( x ) , S ⁡ ( x ) , x = 0 ⁢ ( .2 ) ⁢ 10 ⁢ ( 2 ) ⁢ 100 ⁢ ( 100 ) ⁢ 500 , 8D.

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  • Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of erf ⁡ z , x [ 0 , 5 ] , y = 0.5 ⁢ ( .5 ) ⁢ 3 , 7D and 8D, respectively; the real and imaginary parts of x e ± i ⁢ t 2 ⁢ d t , ( 1 / π ) ⁢ e ∓ i ⁢ ( x 2 + ( π / 4 ) ) ⁢ x e ± i ⁢ t 2 ⁢ d t , x = 0 ⁢ ( .5 ) ⁢ 20 ⁢ ( 1 ) ⁢ 25 , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.