Here are the updated lecture notes with chapters on Laurent Series and Residue Theorem: Download. I have consulted notes by Kilpeläinen (in Finnish) in preparing these, but there are many new topics. Consult the beautifully written complex analysis 1 notes (in English) by Orponen if you need a brush-up on that material. Specially important notions from CA1 for our purposes will be: Cauchy's integral formula, winding number, Morera's theorem and removability of one singularity, and Schwarz lemma. (The weekly schedule. Download.)

Link to Complex Analysis course material by T. Orponen: https://sites.google.com/view/tuomaths/teaching/complex-analysis-i

A visual guide to Möbius transformations -- uses Riemann sphere too. 

Riemann Mapping Theorem in action: See how mainland Finland deforms into the unit disk while keeping (locally) angles preserved. Tiny squares are kept (almost) squares throughout the deformation. Master's Thesis of Tommi Pettinen at University of Turku.