# convolution integrals

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##### 1: 2.5 Mellin Transform Methods
with $a. One of the two convolution integrals associated with the Mellin transform is of the form
2.5.29 $I(x)=\sum\limits_{j,k=1}^{2}I_{jk}(x),$
2.5.31 $I_{21}(x)=0,$ for $x\geq 1$.
##### 2: 2.6 Distributional Methods
We now derive an asymptotic expansion of $I^{\mu}f(x)$ 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:
2.6.62 $I(x)=\sum_{j=0}^{n-1}a_{j}\mathscr{M}\mskip-3.0muh\mskip 3.0mu\left(j+\alpha% \right)x^{-j-\alpha}+\sum_{k=0}^{n-1}b_{k}\mathscr{M}\mskip-3.0muf\mskip 3.0mu% \left(1-k-\beta\right)x^{-k-\beta}+\delta_{n}(x)$
##### 6: 1.14 Integral Transforms
1.14.5 $(f*g)(t)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}f(t-s)g(s)\,\mathrm{d}s.$
1.14.6 $(f*g)(t)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}F(x)G(x){\mathrm{e}}^{-% \mathrm{i}tx}\,\mathrm{d}x,$
1.14.30 $(f*g)(t)=\int^{t}_{0}f(u)g(t-u)\,\mathrm{d}u.$
1.14.39 $(f*g)(x)=\int^{\infty}_{0}f(y)g\left(\frac{x}{y}\right)\frac{\,\mathrm{d}y}{y}.$
1.14.40 $\int^{\infty}_{0}x^{s-1}(f*g)(x)\,\mathrm{d}x=\mathscr{M}\mskip-3.0muf\mskip 3% .0mu\left(s\right)\mathscr{M}\mskip-3.0mug\mskip 3.0mu\left(s\right).$
##### 7: 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 $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)$. … Voigt functions $\mathsf{U}\left(x,t\right)$, $\mathsf{V}\left(x,t\right)$, 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). …
##### 9: Bibliography W
• B. M. Watrasiewicz (1967) Some useful integrals of $\mathrm{Si}(x),$ $\mathrm{Ci}(x)$ 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.