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11: 16.13 Appell Functions
16.13.1 F 1 ( α ; β , β ; γ ; x , y ) = m , n = 0 ( α ) m + n ( β ) m ( β ) n ( γ ) m + n m ! n ! x m y n , max ( | x | , | y | ) < 1 ,
16.13.3 F 3 ( α , α ; β , β ; γ ; x , y ) = m , n = 0 ( α ) m ( α ) n ( β ) m ( β ) n ( γ ) m + n m ! n ! x m y n , max ( | x | , | y | ) < 1 ,
12: 31.8 Solutions via Quadratures
31.8.3 g = 1 2 max ( 2 max 0 k 3 m k , 1 + N - ( 1 + ( - 1 ) N ) ( 1 2 + min 0 k 3 m k ) ) .
13: Bibliography
  • R. W. Abernathy and R. P. Smith (1993) Algorithm 724: Program to calculate F-percentiles. ACM Trans. Math. Software 19 (4), pp. 481–483.
  • A. G. Adams (1969) Algorithm 39: Areas under the normal curve. The Computer Journal 12 (2), pp. 197–198.
  • D. E. Amos (1983b) Algorithm 610. A portable FORTRAN subroutine for derivatives of the psi function. ACM Trans. Math. Software 9 (4), pp. 494–502.
  • D. E. Amos (1980a) Algorithm 556: Exponential integrals. ACM Trans. Math. Software 6 (3), pp. 420–428.
  • W. L. Anderson (1982) Algorithm 588. Fast Hankel transforms using related and lagged convolutions. ACM Trans. Math. Software 8 (4), pp. 369–370.
  • 14: 11.15 Approximations
  • Newman (1984) gives polynomial approximations for H n ( x ) for n = 0 , 1 , 0 x 3 , and rational-fraction approximations for H n ( x ) - Y n ( x ) for n = 0 , 1 , x 3 . The maximum errors do not exceed 1.2×10⁻⁸ for the former and 2.5×10⁻⁸ for the latter.

  • 15: 16.21 Differential Equation
    This equation is of order max ( p , q ) . …
    16: 30.15 Signal Analysis
    The maximum (or least upper bound) B of all numbers …
    30.15.11 arccos B + arccos α = arccos Λ 0 ,
    30.15.12 B = ( Λ 0 α + 1 - Λ 0 1 - α ) 2 .
    17: Bibliography G
  • W. Gautschi (1964b) Algorithm 236: Bessel functions of the first kind. Comm. ACM 7 (8), pp. 479–480.
  • W. Gautschi (1969) Algorithm 363: Complex error function. Comm. ACM 12 (11), pp. 635.
  • W. Gautschi (1973) Algorithm 471: Exponential integrals. Comm. ACM 16 (12), pp. 761–763.
  • W. Gautschi (1977b) Algorithm 521: Repeated integrals of the coerror function. ACM Trans. Math. Software 3, pp. 301–302.
  • A. Gil and J. Segura (1997) Evaluation of Legendre functions of argument greater than one. Comput. Phys. Comm. 105 (2-3), pp. 273–283.
  • 18: 3.2 Linear Algebra
    The p -norm of a vector x = [ x 1 , , x n ] T is given by …
    x = max 1 j n | x j | .
    3.2.14 A p = max x 0 A x p x p .
    A 1 = max 1 k n j = 1 n | a j k | ,
    A = max 1 j n k = 1 n | a j k | ,
    19: 13.10 Integrals
    13.10.3 0 e - z t t b - 1 M ( a , c , k t ) d t = Γ ( b ) z - b F 1 2 ( a , b ; c ; k / z ) , b > 0 , z > max ( k , 0 ) ,
    13.10.7 0 e - z t t b - 1 U ( a , c , t ) d t = Γ ( b ) Γ ( b - c + 1 ) z - b F 1 2 ( a , b ; a + b - c + 1 ; 1 - 1 z ) , b > max ( c - 1 , 0 ) , z > 0 .
    13.10.11 0 t λ - 1 U ( a , b , t ) d t = Γ ( λ ) Γ ( a - λ ) Γ ( λ - b + 1 ) Γ ( a ) Γ ( a - b + 1 ) , max ( b - 1 , 0 ) < λ < a .
    13.10.15 0 t 1 2 ν U ( a , b , t ) J ν ( 2 x t ) d t = Γ ( ν - b + 2 ) Γ ( a ) x 1 2 ν U ( ν - b + 2 , ν - a + 2 , x ) , x > 0 , max ( b - 2 , - 1 ) < ν < 2 a + 1 2 ,
    13.10.16 0 e - t t 1 2 ν U ( a , b , t ) J ν ( 2 x t ) d t = Γ ( ν - b + 2 ) x 1 2 ν e - x M ( a , a - b + ν + 2 , x ) , x > 0 , max ( b - 2 , - 1 ) < ν .
    20: 1.4 Calculus of One Variable
    Maxima and Minima
    as max ( x j + 1 - x j ) 0 . …
    §1.4(vii) Maxima and Minima
    If f ( x ) is twice-differentiable, and if also f ( x 0 ) = 0 and f ′′ ( x 0 ) < 0 ( > 0 ), then x = x 0 is a local maximum (minimum) (§1.4(iii)) of f ( x ) . The overall maximum (minimum) of f ( x ) on [ a , b ] will either be at a local maximum (minimum) or at one of the end points a or b . …