About complex integration.

Question

I was reading a Physics text book and this formula was given without any proof. It seems like a very basic one but I could find information about it anywhere. Not sure if it has a name or not. Given a complex number $\phi_\alpha$, how to derive this equation below: $$ \frac{\mathbb{d}\phi_\alpha^*\cdot\mathbb{d}\phi_\alpha}{2\pi i}=\frac{\mathbb{d}\text{Re}(\phi_\alpha)\cdot \mathbb{d}\text{Im}(\phi_\alpha)}{\pi} $$ Not sure if this help, but I found this in the closure relation of boson coherent state, that is: $$ \int \prod _\alpha{\frac{\mathbb{d}\phi_\alpha^*\cdot \mathbb{d}\phi_\alpha}{2\pi i}}e^{-\sum_\alpha{\phi_\alpha^*\phi_\alpha} } |\phi><\phi|=1 $$ Hope someone could help me explain or point me to some resources that I could read about it.