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21: 23.6 Relations to Other Functions
With z = π u / ( 2 ω 1 ) , …
22: 8.27 Approximations
  • DiDonato (1978) gives a simple approximation for the function F ( p , x ) = x - p e x 2 / 2 x e - t 2 / 2 t p d t (which is related to the incomplete gamma function by a change of variables) for real p and large positive x . This takes the form F ( p , x ) = 4 x / h ( p , x ) , approximately, where h ( p , x ) = 3 ( x 2 - p ) + ( x 2 - p ) 2 + 8 ( x 2 + p ) and is shown to produce an absolute error O ( x - 7 ) as x .

  • 23: 9.5 Integral Representations
    9.5.7 Ai ( z ) = e - ζ π 0 exp ( - z 1 / 2 t 2 ) cos ( 1 3 t 3 ) d t , | ph z | < π .
    9.5.8 Ai ( z ) = e - ζ ζ - 1 / 6 π ( 48 ) 1 / 6 Γ ( 5 6 ) 0 e - t t - 1 / 6 ( 2 + t ζ ) - 1 / 6 d t , | ph z | < 2 3 π .
    24: 27.2 Functions
    Gauss and Legendre conjectured that π ( x ) is asymptotic to x / ln x as x :
    27.2.3 π ( x ) x ln x .
    27.2.4 p n n ln n .
    27.2.14 Λ ( n ) = ln p , n = p a ,
    25: 19.25 Relations to Other Functions
    then the five nontrivial permutations of x , y , z that leave R F invariant change k 2 ( = ( z - y ) / ( z - x ) ) into 1 / k 2 , k 2 , 1 / k 2 , - k 2 / k 2 , - k 2 / k 2 , and sin ϕ ( = ( z - x ) / z ) into k sin ϕ , - i tan ϕ , - i k tan ϕ , ( k sin ϕ ) / 1 - k 2 sin 2 ϕ , - i k sin ϕ / 1 - k 2 sin 2 ϕ . …
    26: Errata
  • Equation (8.12.5)

    To be consistent with the notation used in (8.12.16), Equation (8.12.5) was changed to read

    8.12.5 e ± π i a 2 i sin ( π a ) Q ( - a , z e ± π i ) = ± 1 2 erfc ( ± i η a / 2 ) - i T ( a , η )
  • 27: 2.5 Mellin Transform Methods
    2.5.9 f ( 1 - z ) = π sin ( π z ) , 0 < z < 1 ,
    2.5.10 h ( z ) = 2 z - 1 Γ ( ν + 1 2 z ) Γ 2 ( 1 - 1 2 z ) Γ ( 1 + ν - 1 2 z ) Γ ( z ) π sin ( π z ) , - 2 ν < z < 1 .
    2.5.11 res z = n [ x - z f ( 1 - z ) h ( z ) ] = ( a n ln x + b n ) x - n ,
    28: 1.14 Integral Transforms
    1.14.17 ( f ) ( s ) = f ( s ) = 0 e - s t f ( t ) d t .
    1.14.19 f ( s ) 0 , s .
    If also lim t 0 + f ( t ) / t exists, then … where A p = tan ( 1 2 π / p ) when 1 < p 2 , or cot ( 1 2 π / p ) when p 2 . …
    1.14.46 1 2 π - f ( u ) e i u x d u = - i ( sign x ) f ( x ) ,
    29: 15.12 Asymptotic Approximations
    15.12.5 F ( a + λ , b - λ c ; 1 2 - 1 2 z ) = 2 ( a + b - 1 ) / 2 ( z + 1 ) ( c - a - b - 1 ) / 2 ( z - 1 ) c / 2 ζ sinh ζ ( λ + 1 2 a - 1 2 b ) 1 - c ( I c - 1 ( ( λ + 1 2 a - 1 2 b ) ζ ) ( 1 + O ( λ - 2 ) ) + I c - 2 ( ( λ + 1 2 a - 1 2 b ) ζ ) 2 λ + a - b ( ( c - 1 2 ) ( c - 3 2 ) ( 1 ζ - coth ζ ) + 1 2 ( 2 c - a - b - 1 ) ( a + b - 1 ) tanh ( 1 2 ζ ) + O ( λ - 2 ) ) ) ,
    15.12.9 ( z + 1 ) 3 λ / 2 ( 2 λ ) c - 1 F ( a + λ , b + 2 λ c ; - z ) = λ - 1 / 3 ( e π i ( a - c + λ + ( 1 / 3 ) ) Ai ( e - 2 π i / 3 λ 2 / 3 β 2 ) + e π i ( c - a - λ - ( 1 / 3 ) ) Ai ( e 2 π i / 3 λ 2 / 3 β 2 ) ) ( a 0 ( ζ ) + O ( λ - 1 ) ) + λ - 2 / 3 ( e π i ( a - c + λ + ( 2 / 3 ) ) Ai ( e - 2 π i / 3 λ 2 / 3 β 2 ) + e π i ( c - a - λ - ( 2 / 3 ) ) Ai ( e 2 π i / 3 λ 2 / 3 β 2 ) ) ( a 1 ( ζ ) + O ( λ - 1 ) ) ,
    15.12.11 β = ( - 3 2 ζ + 9 4 ln ( 2 + e ζ 2 + e - ζ ) ) 1 / 3 ,
    15.12.13 G 0 ( ± β ) = ( 2 + e ± ζ ) c - b - ( 1 / 2 ) ( 1 + e ± ζ ) a - c + ( 1 / 2 ) ( z - 1 - e ± ζ ) - a + ( 1 / 2 ) β e ζ - e - ζ .
    30: 19.2 Definitions
    The cases with ϕ = π / 2 are the complete integrals: …
    k c = k ,
    p = 1 - α 2 ,
    x = tan ϕ ,
    If 1 < k 1 / sin ϕ , then k c is pure imaginary. …