About the Project
NIST

double products

AdvancedHelp

(0.001 seconds)

1—10 of 27 matching pages

1: 20.5 Infinite Products and Related Results
§20.5(iii) Double Products
These double products are not absolutely convergent; hence the order of the limits is important. …
2: 23.2 Definitions and Periodic Properties
The double series and double product are absolutely and uniformly convergent in compact sets in that do not include lattice points. …
3: 27.5 Inversion Formulas
27.5.8 g ( n ) = d | n f ( d ) f ( n ) = d | n ( g ( n d ) ) μ ( d ) .
4: 34.4 Definition: 6 j Symbol
The 6 j symbol is defined by the following double sum of products of 3 j symbols: …
5: 10.22 Integrals
Other Double Products
6: 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 ) ) .
7: 16.14 Partial Differential Equations
In addition to the four Appell functions there are 24 other sums of double series that cannot be expressed as a product of two F 1 2 functions, and which satisfy pairs of linear partial differential equations of the second order. …
8: Mathematical Introduction
In addition, there is a comprehensive account of the great variety of analytical methods that are used for deriving and applying the extremely important asymptotic properties of the special functions, including double asymptotic properties (Chapter 2 and §§10.41(iv), 10.41(v)). …

complex plane (excluding infinity).

empty products

unity.

implies.

is equivalent to.

n !!

double factorial: 2 4 6 n if n = 2 , 4 , 6 , ; 1 3 5 n if n = 1 , 3 , 5 , ; 1 if n = 0 , - 1 .

9: Bibliography R
  • Yu. L. Ratis and P. Fernández de Córdoba (1993) A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.
  • W. H. Reid (1995) Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.
  • W. H. Reid (1997a) Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.
  • W. H. Reid (1997b) Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.
  • M. D. Rogers (2005) Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.
  • 10: Bibliography I
  • M. Ikonomou, P. Köhler, and A. F. Jacob (1995) Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines. Z. Angew. Math. Mech. 75 (12), pp. 917–926.
  • IMSL (commercial C, Fortran, and Java libraries) Visual Numerics, Inc..
  • A. Iserles, P. E. Koch, S. P. Nørsett, and J. M. Sanz-Serna (1991) On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65 (2), pp. 151–175.