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1: 30.10 Series and Integrals
For product formulas and convolutions see Connett et al. (1993). …
2: 35.2 Laplace Transform
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
35.2.3 f 1 * f 2 ( T ) = 0 < X < T f 1 ( T - X ) f 2 ( X ) d X .
3: 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 . …
4: 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 …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 …However, the left-hand side can be considered as the convolution of the two distributions associated with the functions t μ - 1 and t - s - α , given by (2.6.12) and (2.6.13). … The method of distributions can be further extended to derive asymptotic expansions for convolution integrals: …
5: 7.21 Physical Applications
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. …
6: 1.14 Integral Transforms
Convolution
For Fourier transforms, the convolution ( f * g ) ( t ) of two functions f ( t ) and g ( t ) defined on ( - , ) is given by …
Convolution
For Laplace transforms, the convolution of two functions f ( t ) and g ( t ) , defined on [ 0 , ) , is …
Convolution
7: 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 .
8: Bibliography G
  • K. Girstmair (1990b) Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.
  • 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: Bibliography W
  • R. Wong (1979) Explicit error terms for asymptotic expansions of Mellin convolutions. J. Math. Anal. Appl. 72 (2), pp. 740–756.
  • 10: Bibliography
  • W. L. Anderson (1982) Algorithm 588. Fast Hankel transforms using related and lagged convolutions. ACM Trans. Math. Software 8 (4), pp. 369–370.