pdf of beta in biparametric canonic form

Question

I am trying to write the pdf of a beta random variable in its biparametric canonic form such as:

Function 1 $$ f_Y(y; \theta, \phi) = exp \{ \phi[y \theta - b(\theta)] + c(y, \phi) \} \mathbb{1}_A(y) $$

My work so far:

First, I use the beta function so facilitate calculation:

$$ B(\alpha, \beta) =\frac{\Gamma(\alpha)\,\Gamma(\beta)}{\Gamma(\alpha+\beta)} $$

than:

function 2 $$ f_Y(y; \alpha, \beta) = exp\{ - log(B(\alpha, \beta)) + (\alpha - 1)log(y) + (\beta-1)log(1-y)\}\mathbb{1}_{[y \in (0,1)]} $$

But I am not seeing what would be $ \phi, \theta,c(y, \phi)$ e $b(\theta)$ in function 2.