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1: 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.

  • 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.

  • Karpov and Čistova (1964) includes D p ( x ) for p = 2 ( .1 ) 0 , ± x = 0 ( .01 ) 5 ; p = 2 ( .05 ) 0 , ± x = 5 ( .01 ) 10 , 6D.

  • Karpov and Čistova (1968) includes e 1 4 x 2 D p ( x ) and e 1 4 x 2 D p ( i x ) for x = 0 ( .01 ) 5 and x 1 = 0(.001 or .0001)5, p = 1 ( .1 ) 1 , 7D or 8S.

  • Zhang and Jin (1996, pp. 455–473) includes U ( ± n 1 2 , x ) , V ( ± n 1 2 , x ) , U ( ± ν 1 2 , x ) , V ( ± ν 1 2 , x ) , and derivatives, ν = n + 1 2 , n = 0 ( 1 ) 10 ( 10 ) 30 , x = 0.5 , 1 , 5 , 10 , 30 , 50 , 8S; W ( a , ± x ) , W ( a , ± x ) , and derivatives, a = h ( 1 ) 5 + h , x = 0.5 , 1 and a = h ( 1 ) 5 + h , x = 5 , h = 0 , 0.5 , 8S. Also, first zeros of U ( a , x ) , V ( a , x ) , and of derivatives, a = 6 ( .5 ) 1 , 6D; first three zeros of W ( a , x ) and of derivative, a = 0 ( .5 ) 4 , 6D; first three zeros of W ( a , ± x ) and of derivative, a = 0.5 ( .5 ) 5.5 , 6D; real and imaginary parts of U ( a , z ) , a = 1.5 ( 1 ) 1.5 , z = x + i y , x = 0.5 , 1 , 5 , 10 , y = 0 ( .5 ) 10 , 8S.

  • 2: 11.8 Analogs to Kelvin Functions
    §11.8 Analogs to Kelvin Functions
    For properties of Struve functions of argument x e ± 3 π i / 4 see McLachlan and Meyers (1936).
    3: 4.16 Elementary Properties
    Table 4.16.2: Trigonometric functions: quarter periods and change of sign.
    x θ 1 2 π ± θ π ± θ 3 2 π ± θ 2 π ± θ
    sin x sin θ cos θ sin θ cos θ ± sin θ
    cos x cos θ sin θ cos θ ± sin θ cos θ
    tan x tan θ cot θ ± tan θ cot θ ± tan θ
    cot x cot θ tan θ ± cot θ tan θ ± cot θ
    4: 28.25 Asymptotic Expansions for Large z
    D 1 ± = 0 ,
    D 0 ± = 1 ,
    The upper signs correspond to M ν ( 3 ) ( z , h ) and the lower signs to M ν ( 4 ) ( z , h ) . The expansion (28.25.1) is valid for M ν ( 3 ) ( z , h ) when …and for M ν ( 4 ) ( z , h ) when …
    5: 36.7 Zeros
    Inside the cusp, that is, for x 2 < 8 | y | 3 / 27 , the zeros form pairs lying in curved rows. … Just outside the cusp, that is, for x 2 > 8 | y | 3 / 27 , there is a single row of zeros on each side. … The zeros are lines in 𝐱 = ( x , y , z ) space where ph Ψ ( E ) ( 𝐱 ) is undetermined. …Near z = z n , and for small x and y , the modulus | Ψ ( E ) ( 𝐱 ) | has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose z and x repeat distances are given by … The zeros of these functions are curves in 𝐱 = ( x , y , z ) space; see Nye (2007) for Φ 3 and Nye (2006) for Φ ( H ) .
    6: 6.4 Analytic Continuation
    6.4.4 Ci ( z e ± π i ) = ± π i + Ci ( z ) ,
    6.4.5 Chi ( z e ± π i ) = ± π i + Chi ( z ) ,
    6.4.6 f ( z e ± π i ) = π e i z f ( z ) ,
    6.4.7 g ( z e ± π i ) = π i e i z + g ( z ) .
    7: 33.8 Continued Fractions
    33.8.2 H ± H ± = c ± i ρ a b 2 ( ρ η ± i ) + ( a + 1 ) ( b + 1 ) 2 ( ρ η ± 2 i ) + ,
    a = 1 + ± i η ,
    b = ± i η ,
    If we denote u = F / F and p + i q = H + / H + , then
    F = ± ( q 1 ( u p ) 2 + q ) 1 / 2 ,
    8: 4.24 Inverse Trigonometric Functions: Further Properties
    4.24.13 Arcsin u ± Arcsin v = Arcsin ( u ( 1 v 2 ) 1 / 2 ± v ( 1 u 2 ) 1 / 2 ) ,
    4.24.14 Arccos u ± Arccos v = Arccos ( u v ( ( 1 u 2 ) ( 1 v 2 ) ) 1 / 2 ) ,
    4.24.15 Arctan u ± Arctan v = Arctan ( u ± v 1 u v ) ,
    4.24.16 Arcsin u ± Arccos v = Arcsin ( u v ± ( ( 1 u 2 ) ( 1 v 2 ) ) 1 / 2 ) = Arccos ( v ( 1 u 2 ) 1 / 2 u ( 1 v 2 ) 1 / 2 ) ,
    4.24.17 Arctan u ± Arccot v = Arctan ( u v ± 1 v u ) = Arccot ( v u u v ± 1 ) .
    9: 10.34 Analytic Continuation
    10.34.3 I ν ( z e m π i ) = ( i / π ) ( ± e m ν π i K ν ( z e ± π i ) e ( m 1 ) ν π i K ν ( z ) ) ,
    10.34.4 K ν ( z e m π i ) = csc ( ν π ) ( ± sin ( m ν π ) K ν ( z e ± π i ) sin ( ( m 1 ) ν π ) K ν ( z ) ) .
    10.34.6 K n ( z e m π i ) = ± ( 1 ) n ( m 1 ) m K n ( z e ± π i ) ( 1 ) n m ( m 1 ) K n ( z ) .
    I ν ( z ¯ ) = I ν ( z ) ¯ ,
    For complex ν replace ν by ν ¯ on the right-hand sides.
    10: 10.36 Other Differential Equations
    The quantity λ 2 in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by λ 2 if at the same time the symbol 𝒞 in the given solutions is replaced by 𝒵 . …
    10.36.1 z 2 ( z 2 + ν 2 ) w ′′ + z ( z 2 + 3 ν 2 ) w ( ( z 2 + ν 2 ) 2 + z 2 ν 2 ) w = 0 , w = 𝒵 ν ( z ) ,
    10.36.2 z 2 w ′′ + z ( 1 ± 2 z ) w + ( ± z ν 2 ) w = 0 , w = e z 𝒵 ν ( z ) .