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 $$\varphi(s)$$ is analytic in the strip $$a < \Re(s) < b$$, and if it tends to zero uniformly as $$  \Im(s) \to \pm \infty  $$ for any real value c between a and b, with its integral along such a line converging absolutely, then if


 * $$f(x)= \{ \mathcal{M}^{-1} \varphi \} = \frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} x^{-s} \varphi(s)\, ds$$

we have that


 * $$\varphi(s)= \{ \mathcal{M} f \} = \int_0^{\infty} x^{s-1} f(x)\,dx.$$

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


 * $$\varphi(s)=\int_0^{\infty} x^{s-1} f(x)\,dx$$

is absolutely convergent when $$a < \Re(s) < b$$. Then $$f$$ is recoverable via the inverse Mellin transform from its Mellin transform $$\varphi$$. 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 $$\varphi(s)$$ can be strengthened if $$f(x)$$ is continuous. If $$\varphi(s)$$ is analytic in the strip $$a < \Re(s) < b$$, and if $$|\varphi(s)| < K |s|^{-2}$$, 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 $$\varphi$$ for at least $$a < \Re(s) < b$$.

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 $$\varphi$$ to simply make it of polynomial growth in any closed strip contained in the open strip $$a < \Re(s) < b$$.

We may also define a Banach space version of this theorem. If we call by $$L_{\nu, p}(R^{+})$$ the weighted Lp space of complex valued functions $$f$$ on the positive reals such that


 * $$\|f\| = \left(\int_0^\infty |x^\nu f(x)|^p\, \frac{dx}{x}\right)^{1/p} < \infty$$

where ν and p are fixed real numbers with $$p>1$$, then if $$f(x)$$ is in $$L_{\nu, p}(R^{+})$$ with $$1 < p \le 2$$, then $$\varphi(s)$$ belongs to $$L_{\nu, q}(R^{+})$$ with $$q = p/(p-1)$$ and


 * $$f(x)=\frac{1}{2 \pi i} \int_{\nu-i \infty}^{\nu+i \infty} x^{-s} \varphi(s)\,ds.$$

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

Since the two-sided Laplace transform can be defined as


 * $$ \left\{\mathcal{B} f\right\}(s) = \left\{\mathcal{M} f(- \ln x) \right\}(s)$$

these theorems can be immediately applied to it also.