Mellin Transform

The Mellin transform is the integral transform defined by

phi(z) = [画像:int_0^inftyt^(z-1)f(t)dt]

(1)

f(t) = [画像: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

[画像: 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)/z a>0,z<0

H(a-t) (a^z)/z a>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

sint Gamma(z)sin(1/2piz) -1<R[z]<1

cost Gamma(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 [画像:(pisec(1/2piz))/(2z)] 0<R[z]<1

erfct [画像:(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) = [画像:lim_(t->infty)1/(2pii)int_(2-iT)^(2+iT)(x^s)/slnzeta(s)ds]

(4)

[画像:(lnzeta(s))/s] = [画像: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).

Explore with Wolfram|Alpha

WolframAlpha

More things to try:

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.

Referenced on Wolfram|Alpha

Mellin Transform

Cite this as:

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

Mellin Transform

See also

Explore with Wolfram|Alpha

References

Referenced on Wolfram|Alpha

Cite this as:

Subject classifications