Mellin Transform

Question

What is the inverse Mellin transform of (s-1/2)^k on the vertical line Re(s)=a where 0 < a <1 and k is a natural number?

Answer 1

Maple command $with(inttrans):invmellin((s-1/2)^k,s,z,0..1)$ (see http://www.maplesoft.com/support/help/Maple/view.aspx?path=inttrans/invmellin for info) calculates it for concrete values of $k.$ For example, Maple produces $$ -1/32\,{\it Dirac} \left( z-1 \right) -{\frac {121}{16}}\,z{\it Dirac} \left( 1,z-1 \right) -{\frac {165}{4}}\,{\it Dirac} \left( 2,z-1 \right) {z}^{2}-$$ $${\frac {85}{2}}\,{\it Dirac} \left( 3,z-1 \right) {z} ^{3}-{\frac {25}{2}}\,{\it Dirac} \left( 4,z-1 \right) {z}^{4}-{\it Dirac} \left( 5,z-1 \right) {z}^{5} $$ in the case $k=5.$

Answer 2

In fact I was aiming at evaluating the expression:

$I_k=\frac{1}{2\pi i}\int_{a-i\infty}^{a+i\infty} (s-\frac{1}{2})^k x^{s-1} ds, 0 < a < 1, k\in N $ changing the variable $s-\frac{1}{2}=z$

one can rewrite the integral $I_k$ as:

$I_k=\frac{1}{2\pi i}\int_{a-\frac{1}{2}-i\infty}^{a-\frac{1}{2}+i\infty} z^k x^{z-\frac{1}{2}} dz$

Recall that the Dirac delta function is defined in the integral form as:

$\delta(u)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{ipu} dp$

One could rewrite it as:

$\delta(u)=\frac{e^{-cu}}{2\pi i}\int_{c-i\infty}^{c+i\infty} e^{zu}dz , Re(z)=c$

which would imply that:

${e^{cu}}\delta(u)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} e^{zu}dz , Re(z)=c$

this in turn yields:

$\frac{d^k}{d u^k}{e^{cu}}\delta(u)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} z^k e^{zu}dz , Re(z)=c, k\in N$

Now we could express the integral $I_k$ as a Fourier transform by making the change of variable $x=e^u$, hence:

$I_k=\frac{1}{2\pi i}\int_{a-\frac{1}{2}-i\infty}^{a-\frac{1}{2}+i\infty} z^k e^{u(z-\frac{1}{2})} dz$

which is the same as: $e^{u/2}I_k=\frac{1}{2\pi i}\int_{a-\frac{1}{2}-i\infty}^{a-\frac{1}{2}+i\infty} z^k e^{zu} dz$

set $c=a-\frac{1}{2}$ then:

$e^{u/2}I_k=\frac{d^k}{d u^k}{e^{cu}}\delta(u)$

Going to our old variable $x$ the above expression becomes:

$\sqrt x\ I_k=\frac{d^k}{d (ln(x))^k}{x^{c}}\delta(ln(x))$

Since $ln(1)=0$ then $f(x)=ln(x)$ has a root at $x=1$. This in turn would imply:

$\delta(ln(x))=\delta(f(x))=\frac{\delta(x-1)}{|f'(1)|}=\delta(x-1)$

Also $\frac{d}{d ln(x)}=x\frac{d}{dx}\rightarrow \frac{d^k}{d (ln(x))^k}=[x\frac{d}{dx}]^k$

Thus finally we get that:

$\sqrt x\ I_k=[x\frac{d}{dx}]^k(x^{c}\delta(x-1))\leftrightarrow I_k=\frac{1}{\sqrt x}[x\frac{d}{dx}]^k(x^{c}\delta(x-1))$

I hope this is a correct solution as it is an important step in a work that I have to complete. Any suggestions if this happens to be wrong would be more than appreciated.