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 φ ( s ) {\displaystyle \varphi (s)} is analytic in the strip a < ℜ ( s ) < b {\displaystyle a<\Re (s)<b} , and if it tends to zero uniformly as ℑ ( s ) → ± ∞ {\displaystyle \Im (s)\to \pm \infty } for any real value c {\displaystyle c} between a {\displaystyle a} and b {\displaystyle b} , with its integral along such a line converging absolutely, then if
f ( x ) = { M − 1 φ } = 1 2 π i ∫ c − i ∞ c + i ∞ x − s φ ( s ) d s {\displaystyle f(x)=\{{\mathcal {M}}^{-1}\varphi \}={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }x^{-s}\varphi (s)\,ds}
we have that
φ ( s ) = { M f } = ∫ 0 ∞ x s − 1 f ( x ) d x . {\displaystyle \varphi (s)=\{{\mathcal {M}}f\}=\int _{0}^{\infty }x^{s-1}f(x)\,dx.}
Conversely, suppose f ( x ) {\displaystyle 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
φ ( s ) = ∫ 0 ∞ x s − 1 f ( x ) d x {\displaystyle \varphi (s)=\int _{0}^{\infty }x^{s-1}f(x)\,dx}
is absolutely convergent when a < ℜ ( s ) < b {\displaystyle a<\Re (s)<b} . Then f {\displaystyle f} is recoverable via the inverse Mellin transform from its Mellin transform φ {\displaystyle \varphi } . These results can be obtained by relating the Mellin transform to the Fourier transform by a change of variables and then applying an appropriate version of the Fourier inversion theorem.
Boundedness condition The boundedness condition on φ ( s ) {\displaystyle \varphi (s)} can be strengthened if f ( x ) {\displaystyle f(x)} is continuous. If φ ( s ) {\displaystyle \varphi (s)} is analytic in the strip a < ℜ ( s ) < b {\displaystyle a<\Re (s)<b} , and if | φ ( s ) | < K | s | − 2 {\displaystyle \vert \varphi (s)\vert <K\vert s\vert ^{-2}} , where K {\displaystyle K} is a positive constant, then f ( x ) {\displaystyle f(x)} as defined by the inversion integral exists and is continuous; moreover the Mellin transform of f {\displaystyle f} is φ {\displaystyle \varphi } for at least a < ℜ ( s ) < b {\displaystyle a<\Re (s)<b} . On the other hand, if we are willing to accept an original f {\displaystyle f} which is a generalized function, we may relax the boundedness condition on φ {\displaystyle \varphi } to simply make it of polynomial growth in any closed strip contained in the open strip a < ℜ ( s ) < b {\displaystyle a<\Re (s)<b} . We may also define a Banach space version of this theorem. If we call by L ν , p ( R + ) {\displaystyle L_{\nu ,p}(R^{+})} the weighted Lp space of complex valued functions f {\displaystyle f} on the positive reals such that
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