Mellin inversion theorem

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In mathematics, the Mellin inversion formula (named after Hjalmar Mellin) tells us conditions under which the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function.

Method[edit]

If is analytic in the strip , and if it tends to zero uniformly as for any real value c between a and b, with its integral along such a line converging absolutely, then if

we have that

Conversely, suppose f(x) is piecewise continuous on the positive real numbers, taking a value halfway between the limit values at any jump discontinuities, and suppose the integral

is absolutely convergent when . Then f is recoverable via the inverse Mellin transform from its Mellin transform [citation needed].

Boundedness condition[edit]

We may strengthen the boundedness condition on if f(x) is continuous. If is analytic in the strip , and if , where K is a positive constant, then f(x) as defined by the inversion integral exists and is continuous; moreover the Mellin transform of f is for at least .

On the other hand, if we are willing to accept an original f which is a generalized function, we may relax the boundedness condition on to simply make it of polynomial growth in any closed strip contained in the open strip .

We may also define a Banach space version of this theorem. If we call by the weighted Lp space of complex valued functions f on the positive reals such that

where ν and p are fixed real numbers with p>1, then if f(x) is in with , then belongs to with and

Here functions, identical everywhere except on a set of measure zero, are identified.

Since the two-sided Laplace transform can be defined as

these theorems can be immediately applied to it also.

See also[edit]

References[edit]

  • P. Flajolet, X. Gourdon, P. Dumas, Mellin transforms and asymptotics: Harmonic sums, Theoretical Computer Science, 144(1-2):3-58, June 1995
  • McLachlan, N. W., Complex Variable Theory and Transform Calculus, Cambridge University Press, 1953.
  • Polyanin, A. D. and Manzhirov, A. V., Handbook of Integral Equations, CRC Press, Boca Raton, 1998.
  • Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals, Oxford University Press, second edition, 1948.
  • Yakubovich, S. B., Index Transforms, World Scientific, 1996.
  • Zemanian, A. H., Generalized Integral Transforms, John Wiley & Sons, 1968.

External links[edit]