  The energy levels of hydrogen-like atoms are obtained from the phase-space
quantization, one of the pillars of the old quantum theory, by three different
methods - (i) direct integration, (ii) Sommerfeld's original method, and (iii)
complex integration. The difficulties come from the imposition of elliptical
orbits to the electron, resulting in a variable radial component of the linear
momentum. Details of the calculation, which constitute a recurrent gap in
textbooks that deal with phase-space quantization, are shown in depth in an
accessible fashion for students of introductory quantum mechanics courses.
