Let's consider beta distribution $B(\theta, 1)$ with density: $$f(x, \theta) = \theta x^{\theta - 1} \cdot 1_{(0, 1)}$$
I want to find a posteriori distribution of $\theta$ when gamma distibution was chosen as prori:
$$\pi(\theta) = \frac{1}{\Gamma(a)b^a}\theta^{a - 1}e^{-\frac{\theta}{b}}$$
My work so far
From the bayesian inference we know that
$$P(H|E) = \frac{P(E|H)P(H)}{P(E)}\propto P(E|H)P(H)$$
where $P(E|H)$ is likelihood and $P(H)$ is priori distrubtion.
Let's rewrite above formula:
$$P(E|H)P(H) = \frac{\theta^n(x_1x_2,...,x_n)^{\theta-1}}{\Gamma(a)b^a} \cdot \theta^{a-1}e^{-\frac{\theta}{b}} \prod_{i=1}^n1_{(0, 1)}(x_i)$$
and here I kinda get stuck. I'm not sure how to rewrite those formulas to get another distribution. For me it doesn't look like another beta distribution or gamma distribution. Could you please give me a hand in which direction should I follow?