Onsager's semi-classical quantisation of cyclotron orbits

I have been working through a really nice paper
Topological Berry phase and semiclassical quantization of cyclotron orbits for two dimensional electrons in coupled band models
by J. N. Fuchs, F. Piéchon, M. O. Goerbig and G. Montambaux

The Appendix below I found particularly illuminating

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