Black hole X-ray spectra: notes on the relativistic calculations

This is a collection of notes to calculate electromagnetic spectra of geometrically thin and optically thick accretion disks around black holes. The presentation is intentionally pedagogical and most calculations are reported step by step. In the disk-corona model, the spectrum of a source has three components: a thermal component from the disk, a Comptonized component from the corona, and a reflection component from the disk. These notes review only the relativistic calculations. The formulas presented here are valid for stationary, axisymmetric, asymptotically-flat, circular spacetimes, so they can be potentially used for a large class of black hole solutions.