Mellin Inversion Theorem
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Mellin Inversion Theorem

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

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 [].

Boundedness condition

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 ? 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

References

  • Flajolet, P.; Gourdon, X.; Dumas, P. (1995). "Mellin transforms and asymptotics: Harmonic sums". Theoretical Computer Science. 144 (1-2): 3-58. doi:10.1016/0304-3975(95)00002-E.
  • McLachlan, N. W. (1953). Complex Variable Theory and Transform Calculus. Cambridge University Press.
  • Polyanin, A. D.; Manzhirov, A. V. (1998). Handbook of Integral Equations. Boca Raton: CRC Press. ISBN 0-8493-2876-4.
  • Titchmarsh, E. C. (1948). Introduction to the Theory of Fourier Integrals (Second ed.). Oxford University Press.
  • Yakubovich, S. B. (1996). Index Transforms. World Scientific. ISBN 981-02-2216-5.
  • Zemanian, A. H. (1968). Generalized Integral Transforms. John Wiley & Sons.

External links


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