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Mellin transform
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==Properties== The properties in this table may be found in {{harvtxt|Bracewell|2000}} and {{harvtxt|Erdélyi|1954}}. {| class="wikitable" |+ Properties of the Mellin transform |- ! Function !! Mellin transform !! Fundamental strip !! Comments |- | <math> f(x) </math> | <math> \tilde{f}(s)=\{\mathcal{M}f\}(s)=\int_0^{\infty} f(x) x^s \frac{dx}{x} </math> | <math> \alpha < \Re s < \beta </math> | Definition |- | <math> x^{\nu}\,f(x) </math> | <math> \tilde{f}(s+\nu) </math> | <math> \alpha - \Re \nu < \Re s < \beta - \Re \nu </math> | |- | <math> f(x^{\nu}) </math> | <math> \frac{1}{|\nu|}\,\tilde{f}\left(\frac{s}{\nu}\right) </math> | <math> \alpha < \nu^{-1} \, \Re s < \beta </math> | <math> \nu\in\mathbb{R},\;\nu\neq 0 </math> |- | <math> f(x^{-1}) </math> | <math> \tilde{f}(-s) </math> | <math> -\beta < \Re s < -\alpha </math> | |- | <math> x^{-1}\,f(x^{-1}) </math> | <math> \tilde{f}(1-s) </math> | <math> 1-\beta < \Re s < 1-\alpha </math> | Involution |- | <math> \overline{f(x)} </math> | <math> \overline{\tilde{f}(\overline{s})} </math> | <math> \alpha < \Re s < \beta </math> | Here <math> \overline{z} </math> denotes the complex conjugate of <math>z</math>. |- | <math> f(\nu x) </math> | <math> \nu^{-s} \tilde{f}(s) </math> | <math> \alpha < \Re s < \beta </math> | <math> \nu > 0 </math>, Scaling |- | <math> f(x)\,\ln x </math> | <math> \tilde{f}'(s) </math> | <math> \alpha < \Re s < \beta </math> | |- | <math> f'(x) </math> | <math> -(s-1)\, \tilde{f}(s-1) </math> | <math> \alpha+1 < \Re s < \beta+1 </math> |The domain shift is conditional and requires evaluation against specific convergence behavior. |- | <math> \left( \frac{d}{dx} \right)^n \, f(x) </math> | <math> (-1)^n \, \frac{\Gamma(s)}{\Gamma(s-n)} \tilde{f}(s-n) </math> | <math> \alpha+n < \Re s < \beta+n </math> | |- | <math> x\,f'(x) </math> | <math> - s \,\tilde{f}(s) </math> | <math> \alpha < \Re s < \beta </math> | |- | <math> \left( x\, \frac{d}{dx} \right)^n \, f(x) </math> | <math> (-s)^n \tilde{f}(s) </math> | <math> \alpha < \Re s < \beta </math> | |- | <math> \left( \frac{d}{dx} \, x\right)^n \, f(x) </math> | <math> (1-s)^n \tilde{f}(s) </math> | <math> \alpha < \Re s < \beta </math> | |- | <math> \int_0^x f(y) \, dy </math> | <math> - s^{-1} \,\tilde{f}(s+1) </math> | <math> \alpha-1 < \Re s < \min(\beta-1,0) </math> | Valid only if the integral exists. |- | <math> \int_x^{\infty} f(y) \, dy </math> | <math> s^{-1} \,\tilde{f}(s+1) </math> | <math> \max(\alpha-1,0) < \Re s < \beta-1 </math> | Valid only if the integral exists. |- | <math> \int_0^{\infty} f_1\left(\frac{x}{y}\right) \, f_2(y) \, \frac{dy}{y} </math> | <math> \tilde{f}_1(s) \,\tilde{f}_2(s) </math> | <math> \max(\alpha_1,\alpha_2) < \Re s < \min(\beta_1,\beta_2) </math> | Multiplicative convolution |- | <math> x^{\mu} \int_0^{\infty} y^{\nu} \, f_1\left(\frac{x}{y}\right) \, f_2(y) \, dy </math> | <math> \tilde{f}_1(s+\mu) \,\tilde{f}_2(s+\mu+\nu+1) </math> | | Multiplicative convolution (generalized) |- | <math> x^{\mu} \int_0^{\infty} y^{\nu} \, f_1(x\,y) \, f_2(y) \, dy </math> | <math> \tilde{f}_1(s+\mu) \,\tilde{f}_2(1-s-\mu+\nu) </math> | | Multiplicative convolution (generalized) |- | <math> f_1(x) \, f_2(x) </math> | <math> \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \tilde{f}_1(r)\,\tilde{f}_2(s-r)\,dr </math> | <math> \begin{aligned} \alpha_2+c&<\Re s<\beta_2+c \\ \alpha_1&<c<\beta_1 \end{aligned} </math> | Multiplication. Only valid if integral exists. See Parseval's theorem below for conditions which ensure the existence of the integral. |} ===Parseval's theorem and Plancherel's theorem=== Let <math>f_1(x)</math> and <math>f_2(x)</math> be functions with well-defined Mellin transforms <math>\tilde{f}_{1,2}(s)=\mathcal{M}\{f_{1,2}\}(s)</math> in the fundamental strips <math>\alpha_{1,2}<\real s<\beta_{1,2}</math>. Let <math>c\in\mathbb{R}</math> with <math>\max(\alpha_1,1-\beta_2)<c<\min(\beta_1,1-\alpha_2)</math>. If the functions <math>x^{c-1/2}\,f_1(x)</math> and <math>x^{1/2-c}\,f_2(x)</math> are also square-integrable over the interval <math>(0,\infty)</math>, then [[Parseval's theorem|Parseval's formula]] holds: <ref name="Titchmarsh 1948 95">{{harvtxt|Titchmarsh|1948|p=95}}.</ref> <math display="block"> \int_0^{\infty} f_1(x)\,f_2(x)\,dx = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \tilde{f_1}(s)\,\tilde{f_2}(1-s)\,ds </math> The integration on the right hand side is done along the vertical line <math> \Re r = c</math> that lies entirely within the overlap of the (suitable transformed) fundamental strips. We can replace <math>f_2(x)</math> by <math>f_2(x)\,x^{s_0-1}</math>. This gives following alternative form of the theorem: Let <math>f_1(x)</math> and <math>f_2(x)</math> be functions with well-defined Mellin transforms <math>\tilde{f}_{1,2}(s)=\mathcal{M}\{f_{1,2}\}(s)</math> in the fundamental strips <math>\alpha_{1,2}<\real s<\beta_{1,2}</math>. Let <math>c\in\mathbb{R}</math> with <math> \alpha_1<c<\beta_1 </math> and choose <math>s_0\in\mathbb{C}</math> with <math> \alpha_2< \Re s_0 - c <\beta_2 </math>. If the functions <math>x^{c-1/2}\,f_1(x)</math> and <math>x^{s_0-c-1/2}\,f_2(x)</math> are also square-integrable over the interval <math>(0,\infty)</math>, then we have <ref name="Titchmarsh 1948 95"/> <math display="block"> \int_0^{\infty} f_1(x)\,f_2(x)\,x^{s_0-1}\,dx = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \tilde{f_1}(s)\,\tilde{f_2}(s_0-s)\,ds </math> We can replace <math>f_2(x)</math> by <math>\overline{f_1(x)}</math>. This gives following theorem: Let <math>f(x)</math> be a function with well-defined Mellin transform <math>\tilde{f}(s) = \mathcal{M}\{f\}(s)</math> in the fundamental strip <math>\alpha<\real s<\beta</math>. Let <math>c\in\mathbb{R}</math> with <math>\alpha<c<\beta</math>. If the function <math>x^{c-1/2}\,f(x)</math> is also square-integrable over the interval <math>(0,\infty)</math>, then [[Plancherel theorem|Plancherel's theorem]] holds:<ref>{{harvtxt|Titchmarsh|1948|p=94}}.</ref> <math display="block"> \int_0^{\infty} |f(x)|^2\,x^{2c-1}dx = \frac{1}{2\pi} \int_{-\infty}^{\infty} | \tilde{f}(c+it) |^2 \,dt </math>
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