Mellin transform
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. 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
[1] | 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 |
[2] | H. Mellin, "Ueber den Zusammenhang zwischen linearen Differential- und Differenzengleichungen" Acta Math. , 25 (1902) pp. 139–164 Zbl 32.0348.02 |
[3] | E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948) MR0942661 Zbl 0017.40404 Zbl 63.0367.05 |
[4] | V.A. Ditkin, A.P. Prudnikov, "Transformations intégrales et calcul opérationnel" , MIR (1978) (Translated from Russian) MR0622209 MR0622210 Zbl 0375.44001 |
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 with automorphic functions (cf. 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. [a1]–[a5].
References
[a1] | E. Hecke, "Ueber die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung" Math. Ann. , 112 (1936) pp. 664–699 Zbl 0014.01601 Zbl 62.1207.01 Zbl 63.0264.03 |
[a2] | A. Weil, "Ueber die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung" Math. Ann. , 168 (1967) pp. 149–156 |
[a3] | A. Weil, "Zeta functions and Mellin transforms" , Algebraic geometry (Bombay Coll., 1968) , Oxford Univ. Press & Tata Inst. (1968) pp. 409–426 MR0262247 Zbl 0193.49104 |
[a4] | A. Ogg, "Modular forms and Dirichlet series" , Benjamin (1969) MR0256993 MR0234918 Zbl 0191.38101 |
[a5] | G. Shimura, "Introduction to the arithmetic theory of automorphic functions" , Princeton Univ. Press & Iwanami-Shoten (1971) pp. §3.6, pp 89–94 MR0314766 Zbl 0221.10029 |
Mellin transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mellin_transform&oldid=36080