# integral transforms

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## 1—10 of 113 matching pages

##### 1: 1.14 Integral Transforms
###### §1.14 IntegralTransforms
1.14.32 $\mathscr{M}\left(f\right)\left(s\right)=\mathscr{M}\mskip-3.0mu f\mskip 3.0mu % \left(s\right)=\int^{\infty}_{0}x^{s-1}f(x)\mathrm{d}x.$
If $x^{\sigma-1}f(x)$ is integrable on $(0,\infty)$ for all $\sigma$ in $a<\sigma, then the integral (1.14.32) converges and $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ is an analytic function of $s$ in the vertical strip $a<\Re s. …
###### §1.14(viii) Compendia
For more extensive tables of the integral transforms of this section and tables of other integral transforms, see Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000), Marichev (1983), Oberhettinger (1972, 1974, 1990), Oberhettinger and Badii (1973), Oberhettinger and Higgins (1961), Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).
##### 2: 12.16 Mathematical Applications
PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …Integral transforms and sampling expansions are considered in Jerri (1982).
##### 3: 19.15 Advantages of Symmetry
Symmetry in $x,y,z$ of $R_{F}\left(x,y,z\right)$, $R_{G}\left(x,y,z\right)$, and $R_{J}\left(x,y,z,p\right)$ replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17). …(19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral. …
##### 4: Mourad E. H. Ismail
Ismail serves on several editorial boards including the Cambridge University Press book series Encyclopedia of Mathematics and its Applications, and on the editorial boards of 9 journals including Proceedings of the American Mathematical Society (Integrable Systems and Special Functions Editor); Constructive Approximation; Journal of Approximation Theory; and Integral Transforms and Special Functions. …
##### 6: Nico M. Temme
He has served on the editorial boards of the SIAM Journal on Mathematical Analysis, Mathematics of Computation, ZAMP, and Integral Transforms and Special Functions. …
##### 7: 15.14 Integrals
###### §15.14 Integrals
15.14.1 $\int_{0}^{\infty}x^{s-1}\mathbf{F}\left({a,b\atop c};-x\right)\mathrm{d}x=% \frac{\Gamma\left(s\right)\Gamma\left(a-s\right)\Gamma\left(b-s\right)}{\Gamma% \left(a\right)\Gamma\left(b\right)\Gamma\left(c-s\right)},$ $\min(\Re a,\Re b)>\Re s>0$.
For other integral transforms see Erdélyi et al. (1954b), Prudnikov et al. (1992b, §4.3.43), and also §15.9(ii). …
##### 8: 19.13 Integrals of Elliptic Integrals
###### §19.13(iii) Laplace Transforms
For direct and inverse Laplace transforms for the complete elliptic integrals $K\left(k\right)$, $E\left(k\right)$, and $D\left(k\right)$ see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.
##### 9: 14.31 Other Applications
These functions are also used in the Mehler–Fock integral transform14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). …
##### 10: 13.23 Integrals
###### §13.23(iv) IntegralTransforms in terms of Whittaker Functions
For additional integral transforms see Magnus et al. (1966, p. 189), Prudnikov et al. (1992b, §§4.3.39–4.3.42), and Wimp (1964). …