(0:00) Quiz due by next class period. (0:37) Mathematica project idea (connecting centers of opposing squares constructed on the sides of a quadrilateral generates line segments that are perpendicular). (4:54) The precise connection between complex planar mappings and real planar mappings. (10:21) Linearity for real planar mappings and complex planar mappings. (19:44) The squaring mapping f(z) = z^2, first, for a particular input. (23:34) The real and imaginary parts for f(z) = z^2 and the corresponding real planar mapping. (25:26) Start to explore the mapping properties of f(z) = z^2 by determining the image of the vertical line x = 3. (37:50) Euler's identity derivation via Taylor series centered at zero. (44:52) Derivation using the chain rule and differential equations. (49:56) Defining e^(z) for an arbitrary complex number z. (51:20) Geometric interpretation of the series definition on Mathematica. (56:04) Make sure you know the polar form of complex numbers in terms of the complex exponential, De Moivre's formula, and applications to trigonometric identities and 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.