View source for Mellin transform

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The integral transform

$$M(p)=\int\limits_0^\infty f(t)t^{p-1}dt,\quad p=\sigma+i\tau.$$

The substitution $t=e^{-z}$ reduces it to the [[Laplace transform|Laplace transform]]. The Mellin transform is used for solving a specific class of planar problems for harmonic functions in a sectorial domain, of problems in elasticity theory, etc.

The inversion theorem. Suppose that $\tau^{\sigma-1}f(\tau)\in L(0,\infty)$ and that the function $f(\tau)$ has bounded variation in a neighbourhood of the point $\tau=t$. Then

$$\frac{f(t+0)-f(t-0)}{2}=\frac{1}{2\pi i}\lim_{\lambda\to\infty}\int\limits_{\sigma-i\lambda}^{\sigma+i\lambda}M(s)t^{-s}ds.$$

The representation theorem. Suppose that the function $M(\tau+iu)$ is summable with respect to $u$ on $(-\infty,+\infty)$ and has bounded variation in a neighbourhood of the point $u=t$. Then

$$\frac{M(\sigma+i(t+0))+M(\sigma+i(t-0))}{2}=\lim_{\lambda\to\infty}\int\limits_{1/\lambda}^\lambda f(x)x^{\sigma+it-1}dx,$$

where

$$f(x)=\frac{1}{2\pi i}\int\limits_{\sigma-i\infty}^{\sigma+i\infty}M(s)x^{-s}ds.$$

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Mellin, "Ueber die fundamentelle Wichtigkeit des Satzes von Cauchy für die Theorie der Gamma- und hypergeometrischen Funktionen" ''Acta Soc. Sci. Fennica'' , '''21''' : 1 (1896) pp. 1–115</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Mellin, "Ueber den Zusammenhang zwischen linearen Differential- und Differenzengleichungen" ''Acta Math.'' , '''25''' (1902) pp. 139–164 {{MR|}} {{ZBL|32.0348.02}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948) {{MR|0942661}} {{ZBL|0017.40404}} {{ZBL|63.0367.05}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> V.A. Ditkin, A.P. Prudnikov, "Transformations intégrales et calcul opérationnel" , MIR (1978) (Translated from Russian) {{MR|0622209}} {{MR|0622210}} {{ZBL|0375.44001}} </TD></TR></table>



====Comments====
If $M(p)$ denotes the Mellin transform of $f(t)$, then the [[Parseval equality]] takes the form:

$$\int\limits_0^\infty|f(t)|^2x^{2k-1}dx=\frac{1}{2\pi}\int\limits_{-\infty}^{+\infty}|M(k+iy)|^2dy$$

if $f(t)t^{k-1/2}\in L_2(0,\infty)$.

The Mellin transform also serves to link [[Dirichlet series|Dirichlet series]] with automorphic functions (cf. [[Automorphic function|Automorphic function]]); in particular, the inversion formula plays a role in the proof of a functional equation for Dirichlet series similar to that for the Riemann zeta-function. Cf. [[#References|[a1]]]–[[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Hecke, "Ueber die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung" ''Math. Ann.'' , '''112''' (1936) pp. 664–699 {{MR|}} {{ZBL|0014.01601}} {{ZBL|62.1207.01}} {{ZBL|63.0264.03}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Weil, "Ueber die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung" ''Math. Ann.'' , '''168''' (1967) pp. 149–156</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Weil, "Zeta functions and Mellin transforms" , ''Algebraic geometry (Bombay Coll., 1968)'' , Oxford Univ. Press &amp; Tata Inst. (1968) pp. 409–426 {{MR|0262247}} {{ZBL|0193.49104}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> A. Ogg, "Modular forms and Dirichlet series" , Benjamin (1969) {{MR|0256993}} {{MR|0234918}} {{ZBL|0191.38101}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> G. Shimura, "Introduction to the arithmetic theory of automorphic functions" , Princeton Univ. Press &amp; Iwanami-Shoten (1971) pp. §3.6, pp 89–94 {{MR|0314766}} {{ZBL|0221.10029}} </TD></TR></table>