(0:00) Lecture plan and the coming weeks. (1:33) Taylor series for f(z) = z/(z^2 + z -12) centered at z = 0 (which will also be the Laurent series centered at the same point) via partial fraction expansion and addition of Taylor series. (16:50) Check answer with "Series" on Mathematica. (24:30) Find the Laurent series centered at z = 0 in the annulus between the circles of radius 3 and 4. (31:43) Use Plot3D and Manipulate on Mathematica to visualize the modulus of f(z) and the modulus of partial sums of the Laurent series. (49:48) The order of a pole. We will relate these topics to integrals.
Based on "Fundamentals of Complex Analysis, with Applications to Engineering and Science", by E.B. Saff and A.D. Snider (3rd Edition). "Visual Complex Analysis", by Tristan Needham, is also referred to a lot. Mathematica is often used, especially to visualize complex analytic (conformal) mappings.