MY MIND IS BLOWN! Istg everything in complex analysis comes from the fact that we study holomorphic functions and god are they beautiful. Holomorphicity implies Cauchy Goursat. Cauchy Coursat leads to the Cauchy integral theorem. The Cauchy integral theorem leads to the generalized Cauchy Integral Theorem. That in turn leads lets us prove that all bounded holomorphic functions are constant. Finally letting us prove by contradictory the fundamental theorem of algebra!
Its like watching a rube goldberg machine or pure beauty. Every small step leads to another step and ends up yielding more and more beautiful results from the single idea of complex differentiability. I cant wait to learn about the residue theorem next week in class!