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 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 .