About the Project
NIST

double

AdvancedHelp

(0.000 seconds)

1—10 of 91 matching pages

1: 10.53 Power Series
10.53.1 j n ( z ) = z n k = 0 ( - 1 2 z 2 ) k k ! ( 2 n + 2 k + 1 ) !! ,
10.53.2 y n ( z ) = - 1 z n + 1 k = 0 n ( 2 n - 2 k - 1 ) !! ( 1 2 z 2 ) k k ! + ( - 1 ) n + 1 z n + 1 k = n + 1 ( - 1 2 z 2 ) k k ! ( 2 k - 2 n - 1 ) !! .
10.53.3 i n ( 1 ) ( z ) = z n k = 0 ( 1 2 z 2 ) k k ! ( 2 n + 2 k + 1 ) !! ,
10.53.4 i n ( 2 ) ( z ) = ( - 1 ) n z n + 1 k = 0 n ( 2 n - 2 k - 1 ) !! ( - 1 2 z 2 ) k k ! + 1 z n + 1 k = n + 1 ( 1 2 z 2 ) k k ! ( 2 k - 2 n - 1 ) !! .
2: Bibliography Q
  • F. Qi (2008) A new lower bound in the second Kershaw’s double inequality. J. Comput. Appl. Math. 214 (2), pp. 610–616.
  • W.-Y. Qiu and R. Wong (2000) Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.
  • 3: Gerhard Wolf
     Schmidt) of the Chapter Double Confluent Heun Equation in the book Heun’s Differential Equations (A. …
    4: 10.72 Mathematical Applications
    If f ( z ) has a double zero z 0 , or more generally z 0 is a zero of order m , m = 2 , 3 , 4 , , then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order 1 / ( m + 2 ) . The number m can also be replaced by any real constant λ ( > - 2 ) in the sense that ( z - z 0 ) - λ f ( z ) is analytic and nonvanishing at z 0 ; moreover, g ( z ) is permitted to have a single or double pole at z 0 . The order of the approximating Bessel functions, or modified Bessel functions, is 1 / ( λ + 2 ) , except in the case when g ( z ) has a double pole at z 0 . …
    §10.72(iii) Differential Equations with a Double Pole and a Movable Turning Point
    In (10.72.1) assume f ( z ) = f ( z , α ) and g ( z ) = g ( z , α ) depend continuously on a real parameter α , f ( z , α ) has a simple zero z = z 0 ( α ) and a double pole z = 0 , except for a critical value α = a , where z 0 ( a ) = 0 . …
    5: 27.5 Inversion Formulas
    27.5.3 g ( n ) = d | n f ( d ) f ( n ) = d | n g ( d ) μ ( n d ) .
    27.5.4 n = d | n ϕ ( d ) ϕ ( n ) = d | n d μ ( n d ) ,
    27.5.6 G ( x ) = n x F ( x n ) F ( x ) = n x μ ( n ) G ( x n ) ,
    27.5.7 G ( x ) = m = 1 F ( m x ) m s F ( x ) = m = 1 μ ( m ) G ( m x ) m s ,
    27.5.8 g ( n ) = d | n f ( d ) f ( n ) = d | n ( g ( n d ) ) μ ( d ) .
    6: 5.17 Barnes’ G -Function (Double Gamma Function)
    §5.17 Barnes’ G -Function (Double Gamma Function)
    5.17.2 G ( n ) = ( n - 2 ) ! ( n - 3 ) ! 1 ! , n = 2 , 3 , .
    5.17.3 G ( z + 1 ) = ( 2 π ) z / 2 exp ( - 1 2 z ( z + 1 ) - 1 2 γ z 2 ) k = 1 ( ( 1 + z k ) k exp ( - z + z 2 2 k ) ) .
    5.17.4 Ln G ( z + 1 ) = 1 2 z ln ( 2 π ) - 1 2 z ( z + 1 ) + z Ln Γ ( z + 1 ) - 0 z Ln Γ ( t + 1 ) d t .
    5.17.5 Ln G ( z + 1 ) 1 4 z 2 + z Ln Γ ( z + 1 ) - ( 1 2 z ( z + 1 ) + 1 12 ) Ln z - ln A + k = 1 B 2 k + 2 2 k ( 2 k + 1 ) ( 2 k + 2 ) z 2 k .
    7: 5.1 Special Notation
    8: 31.9 Orthogonality
    31.9.2 ζ ( 1 + , 0 + , 1 - , 0 - ) t γ - 1 ( 1 - t ) δ - 1 ( t - a ) ϵ - 1 w m ( t ) w k ( t ) d t = δ m , k θ m .
    The integration path is called a Pochhammer double-loop contour (compare Figure 5.12.3). …
    §31.9(ii) Double Orthogonality
    and the integration paths 1 , 2 are Pochhammer double-loop contours encircling distinct pairs of singularities { 0 , 1 } , { 0 , a } , { 1 , a } . …
    9: 8.13 Zeros
    As x increases the positive zeros coalesce to form a double zero at ( a n * , x n * ). The values of the first six double zeros are given to 5D in Table 8.13.1. …
    Table 8.13.1: Double zeros ( a n * , x n * ) of γ * ( a , x ) .
    n a n * x n *
    10: 35.10 Methods of Computation
    Other methods include numerical quadrature applied to double and multiple integral representations. …