Mellin transform

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1: 1.14 Integral Transforms
§1.14(iv) MellinTransform
1.14.32 $\mathscr{M}\left(f\right)\left(s\right)=\mathscr{M}\mskip-3.0mu f\mskip 3.0mu % \left(s\right)=\int^{\infty}_{0}x^{s-1}f(x)\mathrm{d}x.$
2: 2.5 Mellin Transform Methods
2.5.4 $\mathscr{M}\mskip-3.0mu I\mskip 3.0mu \left(z\right)=\mathscr{M}\mskip-3.0mu f% \mskip 3.0mu \left(1-z\right)\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z% \right).$
To apply the Mellin transform method outlined in §2.5(i), we require the transforms $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)$ and $\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)$ to have a common strip of analyticity. …
2.5.26 $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)=\mathscr{M}\mskip-3.0mu f% _{1}\mskip 3.0mu \left(z\right)+\mathscr{M}\mskip-3.0mu f_{2}\mskip 3.0mu % \left(z\right)$
2.5.28 $\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)=\mathscr{M}\mskip-3.0mu h% _{1}\mskip 3.0mu \left(z\right)+\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu % \left(z\right)$
3: 2.6 Distributional Methods
2.6.18 $c_{s}=\frac{(-1)^{s}}{(s-1)!}\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(s% \right),$
$\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ being the Mellin transform of $f(t)$ or its analytic continuation (§2.5(ii)). …
2.6.60 $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)=\mathscr{M}\mskip-3.0mu f% _{n}\mskip 3.0mu \left(z\right),$
where $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ is the Mellin transform of $f$ or its analytic continuation. …
2.6.61 $\mathscr{M}\mskip-3.0mu h_{x}\mskip 3.0mu \left(j+\alpha\right)=x^{-j-\alpha}% \mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(j+\alpha\right),$
4: 15.14 Integrals
§15.14 Integrals
The Mellin transform of the hypergeometric function of negative argument is given by
15.14.1 $\int_{0}^{\infty}x^{s-1}\mathbf{F}\left({a,b\atop c};-x\right)\mathrm{d}x=% \frac{\Gamma\left(s\right)\Gamma\left(a-s\right)\Gamma\left(b-s\right)}{\Gamma% \left(a\right)\Gamma\left(b\right)\Gamma\left(c-s\right)},$ $\min(\Re a,\Re b)>\Re s>0$.
Mellin transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §6.9), Oberhettinger (1974, §1.15), and Marichev (1983, pp. 288–299). Inverse Mellin transforms are given in Erdélyi et al. (1954a, §7.5). …
5: 20.10 Integrals
§20.10(i) MellinTransforms with respect to the Lattice Parameter
20.10.1 $\int_{0}^{\infty}x^{s-1}\theta_{2}\left(0\middle|ix^{2}\right)\mathrm{d}x=2^{s% }(1-2^{-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$
20.10.2 $\int_{0}^{\infty}x^{s-1}(\theta_{3}\left(0\middle|ix^{2}\right)-1)\mathrm{d}x=% \pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$
20.10.3 $\int_{0}^{\infty}x^{s-1}(1-\theta_{4}\left(0\middle|ix^{2}\right))\mathrm{d}x=% (1-2^{1-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right).$
6: 2.3 Integrals of a Real Variable
2.3.12 $\int_{0}^{\infty}f(xt)q(t)\mathrm{d}t\sim\sum_{s=0}^{\infty}\mathscr{M}\mskip-% 3.0mu f\mskip 3.0mu \left(\frac{s+\lambda}{\mu}\right)\frac{a_{s}}{x^{(s+% \lambda)/\mu}},$ $x\to+\infty$,
where $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(\alpha\right)$ is the Mellin transform of $f(t)$2.5(i)). …
7: 13.23 Integrals
§13.23(i) Laplace and MellinTransforms
13.23.1 $\int_{0}^{\infty}e^{-zt}t^{\nu-1}M_{\kappa,\mu}\left(t\right)\mathrm{d}t=\frac% {\Gamma\left(\mu+\nu+\tfrac{1}{2}\right)}{\left(z+\frac{1}{2}\right)^{\mu+\nu+% \frac{1}{2}}}\*{{}_{2}F_{1}}\left({\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}+\mu+% \nu\atop 1+2\mu};\frac{1}{z+\frac{1}{2}}\right),$ $\Re(\mu+\nu+\tfrac{1}{2})>0$, $\Re z>\tfrac{1}{2}$.
13.23.2 $\int_{0}^{\infty}e^{-zt}t^{\mu-\frac{1}{2}}M_{\kappa,\mu}\left(t\right)\mathrm% {d}t=\Gamma\left(2\mu+1\right)\left(z+\tfrac{1}{2}\right)^{-\kappa-\mu-\frac{1% }{2}}\*\left(z-\tfrac{1}{2}\right)^{\kappa-\mu-\frac{1}{2}},$ $\Re\mu>-\tfrac{1}{2}$, $\Re z>\tfrac{1}{2}$,
13.23.3 $\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\nu-1}% M_{\kappa,\mu}\left(t\right)\mathrm{d}t=\frac{\Gamma\left(\mu+\nu+\frac{1}{2}% \right)\Gamma\left(\kappa-\nu\right)}{\Gamma\left(\frac{1}{2}+\mu+\kappa\right% )\Gamma\left(\frac{1}{2}+\mu-\nu\right)},$ $-\tfrac{1}{2}-\Re\mu<\Re\nu<\Re\kappa$.
For additional Laplace and Mellin transforms see Erdélyi et al. (1954a, §§4.22, 5.20, 6.9, 7.5), Marichev (1983, pp. 283–287), Oberhettinger and Badii (1973, §1.17), Oberhettinger (1974, §§1.13, 2.8), and Prudnikov et al. (1992a, §§3.34, 3.35). …
9: 13.10 Integrals
§13.10(iii) MellinTransforms
13.10.10 $\int_{0}^{\infty}t^{\lambda-1}{\mathbf{M}}\left(a,b,-t\right)\mathrm{d}t=\frac% {\Gamma\left(\lambda\right)\Gamma\left(a-\lambda\right)}{\Gamma\left(a\right)% \Gamma\left(b-\lambda\right)},$ $0<\Re\lambda<\Re a$,
13.10.11 $\int_{0}^{\infty}t^{\lambda-1}U\left(a,b,t\right)\mathrm{d}t=\frac{\Gamma\left% (\lambda\right)\Gamma\left(a-\lambda\right)\Gamma\left(\lambda-b+1\right)}{% \Gamma\left(a\right)\Gamma\left(a-b+1\right)},$ $\max\left(\Re b-1,0\right)<\Re\lambda<\Re a$.
For additional Mellin transforms see Erdélyi et al. (1954a, §§6.9, 7.5), Marichev (1983, pp. 283–287), and Oberhettinger (1974, §§1.13, 2.8). …
10: 16.15 Integral Representations and Integrals
16.15.1 ${F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=\frac{\Gamma\left(% \gamma\right)}{\Gamma\left(\alpha\right)\Gamma\left(\gamma-\alpha\right)}\int_% {0}^{1}\frac{u^{\alpha-1}(1-u)^{\gamma-\alpha-1}}{(1-ux)^{\beta}(1-uy)^{\beta^% {\prime}}}\mathrm{d}u,$ $\Re\alpha>0$, $\Re(\gamma-\alpha)>0$,
16.15.2 ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)=% \frac{\Gamma\left(\gamma\right)\Gamma\left(\gamma^{\prime}\right)}{\Gamma\left% (\beta\right)\Gamma\left(\beta^{\prime}\right)\Gamma\left(\gamma-\beta\right)% \Gamma\left(\gamma^{\prime}-\beta^{\prime}\right)}\int_{0}^{1}\!\!\!\int_{0}^{% 1}\frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u)^{\gamma-\beta-1}(1-v)^{\gamma^{% \prime}-\beta^{\prime}-1}}{(1-ux-vy)^{\alpha}}\mathrm{d}u\mathrm{d}v,$ $\Re\gamma>\Re\beta>0$, $\Re\gamma^{\prime}>\Re\beta^{\prime}>0$,
16.15.3 ${F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x,y\right)=% \frac{\Gamma\left(\gamma\right)}{\Gamma\left(\beta\right)\Gamma\left(\beta^{% \prime}\right)\Gamma\left(\gamma-\beta-\beta^{\prime}\right)}\iint_{\Delta}% \frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u-v)^{\gamma-\beta-\beta^{\prime}-1}}{% (1-ux)^{\alpha}(1-vy)^{\alpha^{\prime}}}\mathrm{d}u\mathrm{d}v,$ $\Re(\gamma-\beta-\beta^{\prime})>0$, $\Re\beta>0$, $\Re\beta^{\prime}>0$,
16.15.4 ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x(1-y),y(1-x)\right)=\frac{% \Gamma\left(\gamma\right)\Gamma\left(\gamma^{\prime}\right)}{\Gamma\left(% \alpha\right)\Gamma\left(\beta\right)\Gamma\left(\gamma-\alpha\right)\Gamma% \left(\gamma^{\prime}-\beta\right)}\int_{0}^{1}\!\!\!\int_{0}^{1}\frac{u^{% \alpha-1}v^{\beta-1}(1-u)^{\gamma-\alpha-1}(1-v)^{\gamma^{\prime}-\beta-1}}{(1% -ux)^{\gamma+\gamma^{\prime}-\alpha-1}(1-vy)^{\gamma+\gamma^{\prime}-\beta-1}(% 1-ux-vy)^{\alpha+\beta-\gamma-\gamma^{\prime}+1}}\mathrm{d}u\mathrm{d}v,$ $\Re\gamma>\Re\alpha>0$, $\Re\gamma^{\prime}>\Re\beta>0$.