About the Project
NIST

convolution integrals

AdvancedHelp

(0.002 seconds)

1—10 of 13 matching pages

1: 2.5 Mellin Transform Methods
with a < c < b . One of the two convolution integrals associated with the Mellin transform is of the form
2.5.3 I ( x ) = 0 f ( t ) h ( x t ) d t , x > 0 ,
2.5.29 I ( x ) = j , k = 1 2 I j k ( x ) ,
2.5.31 I 21 ( x ) = 0 , for x 1 .
2: 2.6 Distributional Methods
We now derive an asymptotic expansion of I μ f ( x ) for large positive values of x . … The replacement of f ( t ) by its asymptotic expansion (2.6.9), followed by term-by-term integration leads to convolution integrals of the form … The method of distributions can be further extended to derive asymptotic expansions for convolution integrals:
2.6.54 I ( x ) = 0 f ( t ) h ( x t ) d t .
2.6.62 I ( x ) = j = 0 n - 1 a j h ( j + α ) x - j - α + k = 0 n - 1 b k f ( 1 - k - β ) x - k - β + δ n ( x )
3: 30.10 Series and Integrals
4: 10.22 Integrals
Convolutions
5: 1.14 Integral Transforms
1.14.5 ( f * g ) ( t ) = 1 2 π - f ( t - s ) g ( s ) d s .
1.14.30 ( f * g ) ( t ) = 0 t f ( u ) g ( t - u ) d u .
1.14.39 ( f * g ) ( x ) = 0 f ( y ) g ( x y ) d y y .
1.14.40 0 x s - 1 ( f * g ) ( x ) d x = f ( s ) g ( s ) .
6: 7.21 Physical Applications
§7.21 Physical Applications
The error functions, Fresnel integrals, and related functions occur in a variety of physical applications. … Carslaw and Jaeger (1959) gives many applications and points out the importance of the repeated integrals of the complementary error function i n erfc ( z ) . … Voigt functions U ( x , t ) , V ( x , t ) , can be regarded as the convolution of a Gaussian and a Lorentzian, and appear when the analysis of light (or particulate) absorption (or emission) involves thermal motion effects. …Dawson’s integral appears in de-convolving even more complex motional effects; see Pratt (2007). …
7: 35.2 Laplace Transform
35.2.1 g ( Z ) = Ω etr ( - Z X ) f ( X ) d X ,
Assume that 𝒮 | g ( Z ) | d V converges, and also that lim U 𝒮 | g ( Z ) | d V = 0 . …where the integral is taken over all Z = U + i V such that U > X 0 and V ranges over 𝒮 .
Convolution Theorem
If g j is the Laplace transform of f j , j = 1 , 2 , then g 1 g 2 is the Laplace transform of the convolution f 1 * f 2 , where …
8: 18.17 Integrals
§18.17 Integrals
§18.17(i) Indefinite Integrals
§18.17(iv) Fractional Integrals
Ultraspherical
§18.17(viii) Other Integrals
9: Bibliography W
  • B. M. Watrasiewicz (1967) Some useful integrals of Si ( x ) , Ci ( x ) and related integrals. Optica Acta 14 (3), pp. 317–322.
  • A. D. Wheelon (1968) Tables of Summable Series and Integrals Involving Bessel Functions. Holden-Day, San Francisco, CA.
  • J. Wimp (1964) A class of integral transforms. Proc. Edinburgh Math. Soc. (2) 14, pp. 33–40.
  • R. Wong (1973b) On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.
  • R. Wong (1979) Explicit error terms for asymptotic expansions of Mellin convolutions. J. Math. Anal. Appl. 72 (2), pp. 740–756.
  • 10: Bibliography G
  • W. Gautschi (1973) Algorithm 471: Exponential integrals. Comm. ACM 16 (12), pp. 761–763.
  • M. Geller and E. W. Ng (1969) A table of integrals of the exponential integral. J. Res. Nat. Bur. Standards Sect. B 73B, pp. 191–210.
  • K. Girstmair (1990b) Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.
  • M. L. Glasser (1976) Definite integrals of the complete elliptic integral K . J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 313–323.
  • 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.