About the Project
NIST

convolution product

AdvancedHelp

(0.001 seconds)

9 matching pages

1: 27.5 Inversion Formulas
27.5.1 h ( n ) = d | n f ( d ) g ( n d ) ,
called the Dirichlet product (or convolution) of f and g . …
2: 30.10 Series and Integrals
For product formulas and convolutions see Connett et al. (1993). …
3: 2.6 Distributional Methods
We now derive an asymptotic expansion of I μ f ( x ) for large positive values of x . In terms of the convolution product
2.6.34 ( f g ) ( x ) = 0 x f ( x - t ) g ( t ) d t
4: Bibliography C
  • W. C. Connett, C. Markett, and A. L. Schwartz (1993) Product formulas and convolutions for angular and radial spheroidal wave functions. Trans. Amer. Math. Soc. 338 (2), pp. 695–710.
  • 5: 2.5 Mellin Transform Methods
    with a < c < b . One of the two convolution integrals associated with the Mellin transform is of the form … In the half-plane z > max ( 0 , - 2 ν ) , the product f ( 1 - z ) h ( z ) has a pole of order two at each positive integer, and …
    2.5.29 I ( x ) = j , k = 1 2 I j k ( x ) ,
    2.5.31 I 21 ( x ) = 0 , for x 1 .
    6: Bibliography L
  • D. R. Lehman, W. C. Parke, and L. C. Maximon (1981) Numerical evaluation of integrals containing a spherical Bessel function by product integration. J. Math. Phys. 22 (7), pp. 1399–1413.
  • X. Li and R. Wong (1994) Error bounds for asymptotic expansions of Laplace convolutions. SIAM J. Math. Anal. 25 (6), pp. 1537–1553.
  • P. Linz and T. E. Kropp (1973) A note on the computation of integrals involving products of trigonometric and Bessel functions. Math. Comp. 27 (124), pp. 871–872.
  • S. K. Lucas (1995) Evaluating infinite integrals involving products of Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 269–282.
  • 7: 10.22 Integrals
    Products
    Products
    Convolutions
    Other Double Products
    Triple Products
    8: Bibliography G
  • K. Girstmair (1990b) Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.
  • E. T. Goodwin (1949a) Recurrence relations for cross-products of Bessel functions. Quart. J. Mech. Appl. Math. 2 (1), pp. 72–74.
  • H. P. W. Gottlieb (1985) On the exceptional zeros of cross-products of derivatives of spherical Bessel functions. Z. Angew. Math. Phys. 36 (3), pp. 491–494.
  • I. S. Gradshteyn and I. M. Ryzhik (2000) Table of Integrals, Series, and Products. 6th edition, Academic Press Inc., San Diego, CA.
  • D. P. Gupta and M. E. Muldoon (2000) Riccati equations and convolution formulae for functions of Rayleigh type. J. Phys. A 33 (7), pp. 1363–1368.
  • 9: 18.17 Integrals
    §18.17(ii) Integral Representations for Products
    Ultraspherical
    Legendre
    For formulas for Jacobi and Laguerre polynomials analogous to (18.17.5) and (18.17.6), see Koornwinder (1974, 1977). …