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Mellin Transform

The Mellin transform is the integral transform defined by

phi(z)=int_0^inftyt^(z-1)f(t)dt
(1)
f(t)=1/(2pii)int_(c-iinfty)^(c+iinfty)t^(-z)phi(z)dz.
(2)

It is implemented in the Wolfram Language as MellinTransform[expr, x, s].

The transform phi(z) exists if the integral

int_0^infty|f(x)|x^(k-1)dx
(3)

is bounded for some k>0, in which case the inverse f(t) exists with c>k. The functions phi(z) and f(t) are called a Mellin transform pair, and either can be computed if the other is known.

The following table gives Mellin transforms of common functions (Bracewell 1999, p. 255). Here, delta is the delta function, H(x) is the Heaviside step function, Gamma(z) is the gamma function, B(z;a,b) is the incomplete beta function, erfcz is the complementary error function erfc, and Si(z) is the sine integral.

f(t)phi(z)convergence
delta(t-a)a^(z-1)
H(t-a)-(a^z)/za>0,z<0
H(a-t)(a^z)/za>0,z>0
t^nH(t-a)-(a^(n+z))/(n+z)a>0,R[z+n]<0
t^nH(a-t)(a^(n+z))/(n+z)a>0,R[n+z]>0
e^(-at)a^(-z)Gamma(z)R[a],R[z]>0
e^(-t^2)1/2Gamma(1/2z)R[z]>0
sintGamma(z)sin(1/2piz)-1<R[z]<1
costGamma(z)cos(1/2piz)0<R[z]<1
1/(1+t)picsc(piz)0<R[z]<1
1/((1+t)^a)(Gamma(a-z)Gamma(z))/(Gamma(a))R[a-z]>0,R[z]>0
1/(1+t^2)1/2picsc(1/2piz)0<R[z]<2
(1-t)^(a-1)H(1-t)(Gamma(a)Gamma(z))/(Gamma(a+z))R[a],R[z]>0
(t-1)^(-a)H(t-1)(Gamma(1-a)Gamma(a-z))/(Gamma(1-x))R[a-z]>0,R[a]<1
ln(1+t)(picsc(piz))/z-1<R[z]<0
1/2pi-tan^(-1)t(pisec(1/2piz))/(2z)0<R[z]<1
erfct(Gamma(1/2(1+z)))/(sqrt(pi)z)R[z]>0
Si(t)-1/zGamma(z)sin(1/2piz)R[z]>-1
(t^a)/(1-t)H(t-a)-B(a^(-1);1-a-z;0)a>1,R[a+z]<1

Another example of a Mellin transform is the relationship between the Riemann function f(x) and the Riemann zeta function zeta(s),

f(x)=lim_(t->infty)1/(2pii)int_(2-iT)^(2+iT)(x^s)/slnzeta(s)ds
(4)
(lnzeta(s))/s=int_1^inftyf(x)x^(-s-1)dx.
(5)

A related pair is used in one proof of the prime number theorem (Titchmarsh 1987, pp. 51-54 and equation 3.7.2).

See also

Fourier Transform, Integral Transform, Strassen Formulas

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 795, 1985.Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 254-257, 1999.Gradshteyn, I. S. and Ryzhik, I. M. "Mellin Transform." §17.41 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1193-1197, 2000.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 469-471, 1953.Oberhettinger, F. Tables of Mellin Transforms. New York: Springer-Verlag, 1974.Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. "Evaluation of Integrals and the Mellin Transform." Itogi Nauki i Tekhniki, Seriya Matemat. Analiz 27, 3-146, 1989.Titchmarsh, E. C. The Theory of the Riemann Zeta Function, 2nd ed. New York: Clarendon Press, 1987.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 567, 1995.

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Mellin Transform

Cite this as:

Weisstein, Eric W. "Mellin Transform." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MellinTransform.html

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