What is the posterior distribution for parameter $b$ with $X \sim Gamma(a,b)$, under the Jeffreys prior? We can assume that $a$ is known.
The Jeffreys prior is the square of the Fisher information of $b$:
$p(b)=\frac{\sqrt(a)}{b}$.
Then using Bayes' rule we have
$p(b|x) \propto p(x|b) \,p(b) = \dfrac{b^a}{\Gamma (a)}x^{a-1}e^{-xb}\cdot\frac{\sqrt(a)}{b}$
Next we look for the kernel of a Gamma distribution. But this is where I am stuck.
What is the next step for deriving the posterior distribution for $b$?