So, I'm trying to prove Dirichlet’s class number formula for a real quadratic number field. (Exc. 10.5.9 in Problems in Algebraic Number Theory” Murthy, Esmonde.) We define
$$ D= \left\{ (u,v)\in\mathbb{R}^2 | \begin{array}{l} 0<|u\beta_1+v\beta_2||u\beta_1'+v\beta_2'|<1, \\ 0<\log\left| \frac{u\beta_1+v\beta_2}{\sqrt{|u\beta_1+v\beta_2|}\sqrt{|u\beta_1'+v\beta_2'|}}\right| <\log\varepsilon \end{array} \right\}. $$ Where $\beta_1,\beta_2$ is a integral basis for an ideal $J$ in $O_K$, with $K$ real quadratic number field. We let $\beta’_1,\beta_2’$ be the conjugates respectively. I know that $\beta$ is on the form $a+b\sqrt{d}$ and $\beta’$ is $a-b\sqrt{d}$. I need to find the volume of D i.e.: $$ \text{vol}D = \int_{\mathbb{R}} 1_D dudv $$ I think I need to use change of variable theorem, and maybe with a change of variable like $\Phi: (u_1,u_2)\mapsto (ua_1+va_2, \sqrt{d}(ub_1+vb_2))$. I think that we can argue that $|u\beta_1+v\beta_2||u\beta_1'+v\beta_2'|=|u_1+u_2||u_1-u_2|\overset{?}{=}u_1^2+u_2^2$ (using the transformation $\Phi$). Furter I also think we would be able to argue that $\left| \frac{u\beta_1+v\beta_2}{\sqrt{|u\beta_1+v\beta_2|}\sqrt{|u\beta_1'+v\beta_2'|}}\right| = \left|\sqrt{\frac{u\beta_1+v\beta_2}{u\beta_1’+v\beta_2’}}\right|=\left|\sqrt{\frac{u_1+u_2}{u_1-u_2}}\right|$. (I'm a bit unsure if the calculation respect the numerical value.) I also think that the integral should integrate to something with the term $\log \varepsilon$ in it. Yeah, but I’m struggling on how to do the integral. I hope someone can help.
PS. If you know a similar proof of Dirichlet’s class number formula for real quadratic number field, please point me in that direction.