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