Mellin transform

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1: 2.5 Mellin Transform Methods
2.5.4 $\mathscr{M}\mskip-3.0muI\mskip 3.0mu\left(z\right)=\mathscr{M}\mskip-3.0muf% \mskip 3.0mu\left(1-z\right)\mathscr{M}\mskip-3.0muh\mskip 3.0mu\left(z\right).$
To apply the Mellin transform method outlined in §2.5(i), we require the transforms $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(1-z\right)$ and $\mathscr{M}\mskip-3.0muh\mskip 3.0mu\left(z\right)$ to have a common strip of analyticity. …
2.5.26 $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)=\mathscr{M}\mskip-3.0muf_{1% }\mskip 3.0mu\left(z\right)+\mathscr{M}\mskip-3.0muf_{2}\mskip 3.0mu\left(z\right)$
2.5.28 $\mathscr{M}\mskip-3.0muh\mskip 3.0mu\left(z\right)=\mathscr{M}\mskip-3.0muh_{1% }\mskip 3.0mu\left(z\right)+\mathscr{M}\mskip-3.0muh_{2}\mskip 3.0mu\left(z\right)$
2: 1.14 Integral Transforms
§1.14(iv) MellinTransform
1.14.32 $\mathscr{M}\left(f\right)\left(s\right)=\mathscr{M}\mskip-3.0muf\mskip 3.0mu% \left(s\right)=\int^{\infty}_{0}x^{s-1}f(x)\mathrm{d}x.$
3: 2.6 Distributional Methods
2.6.18 $c_{s}=\frac{(-1)^{s}}{(s-1)!}\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right),$
$\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ being the Mellin transform of $f(t)$ or its analytic continuation (§2.5(ii)). …
2.6.60 $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)=\mathscr{M}\mskip-3.0muf_{n% }\mskip 3.0mu\left(z\right),$
where $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ is the Mellin transform of $f$ or its analytic continuation. …
2.6.61 $\mathscr{M}\mskip-3.0muh_{x}\mskip 3.0mu\left(j+\alpha\right)=x^{-j-\alpha}% \mathscr{M}\mskip-3.0muh\mskip 3.0mu\left(j+\alpha\right),$
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: 2.3 Integrals of a Real Variable
2.3.12 $\int_{0}^{\infty}f(xt)q(t)\mathrm{d}t\sim\sum_{s=0}^{\infty}\mathscr{M}\mskip-% 3.0muf\mskip 3.0mu\left(\frac{s+\lambda}{\mu}\right)\frac{a_{s}}{x^{(s+\lambda% )/\mu}},$ $x\to+\infty$,
where $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(\alpha\right)$ is the Mellin transform of $f(t)$2.5(i)). …
§2.3(vi) Asymptotics of MellinTransforms
For the asymptotics of the Mellin transform $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)=\int^{\infty}_{0}t^{z-1}f(t% )\mathrm{d}t$ as $z\to\infty$ see Frenzen (1987b), Sidi (1985, 2011).
6: 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 $\int_{0}^{\infty}x^{s-1}(1-\theta_{4}\left(0\middle|ix^{2}\right))\mathrm{d}x=% (1-2^{1-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right).$
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 ${F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=\frac{\Gamma\left(% \gamma\right)}{\Gamma\left(\alpha\right)\Gamma\left(\gamma-\alpha\right)}\int_% {0}^{1}\frac{u^{\alpha-1}(1-u)^{\gamma-\alpha-1}}{(1-ux)^{\beta}(1-uy)^{\beta^% {\prime}}}\mathrm{d}u,$ $\Re\alpha>0$, $\Re\left(\gamma-\alpha\right)>0$,
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 ${F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x,y\right)=% \frac{\Gamma\left(\gamma\right)}{\Gamma\left(\beta\right)\Gamma\left(\beta^{% \prime}\right)\Gamma\left(\gamma-\beta-\beta^{\prime}\right)}\iint_{\Delta}% \frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u-v)^{\gamma-\beta-\beta^{\prime}-1}}{% (1-ux)^{\alpha}(1-vy)^{\alpha^{\prime}}}\mathrm{d}u\mathrm{d}v,$ $\Re\left(\gamma-\beta-\beta^{\prime}\right)>0$, $\Re\beta>0$, $\Re\beta^{\prime}>0$,
16.15.4 ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x(1-y),y(1-x)\right)=\frac{% \Gamma\left(\gamma\right)\Gamma\left(\gamma^{\prime}\right)}{\Gamma\left(% \alpha\right)\Gamma\left(\beta\right)\Gamma\left(\gamma-\alpha\right)\Gamma% \left(\gamma^{\prime}-\beta\right)}\int_{0}^{1}\!\!\!\int_{0}^{1}\frac{u^{% \alpha-1}v^{\beta-1}(1-u)^{\gamma-\alpha-1}(1-v)^{\gamma^{\prime}-\beta-1}}{(1% -ux)^{\gamma+\gamma^{\prime}-\alpha-1}(1-vy)^{\gamma+\gamma^{\prime}-\beta-1}(% 1-ux-vy)^{\alpha+\beta-\gamma-\gamma^{\prime}+1}}\mathrm{d}u\mathrm{d}v,$ $\Re\gamma>\Re\alpha>0$, $\Re\gamma^{\prime}>\Re\beta>0$.