de'Broglie's paradox, relativistic Doppler effect, and the derivation of mass-energy relation in special relativity

Three related topics are discussed to show the "simplicity, harmony and beauty" of the theory of special relativity (SR): (1) How can de'Broglie discover his famous relation between the particle's momentum and a wave length -- a paradox stemming from the time dilatation effect of SR. (2) The relativistic Doppler effect, especially the distinction between the transverse Doppler effect and the time dilatation effect. (3) Three particular examples are examined to derive the general mass-energy relation in SR by induction method.