Intro Complex Analysis, Lec 11, Areas of Images, Differentiability, Analyticity, Cauchy-Riemann Eqs

Lecture 11. (0:00) Exam 1 next class period. (2:05) Areas of images of complex mappings (complex function). Use a change of variables for a double integral with the determinant of the Jacobian matrix (the Jacobian determinant). Example: Find the area of the image of the unit disk under the mapping w = f(z) = z^2 + 4z + i. (12:50) Reminder of the definition of an analytic function. (13:40) The conjugate mapping is not differentiable everywhere. The function f(z) = |z|^2 fails to be differentiable except at z = 0. It is therefore nowhere analytic. Verify by thinking about the definition of the derivative (as a limit of a difference quotient). (23:44) Be able to use the limit definition of the derivative to find a derivative and give reasons. (25:00) Motivation and intuition for the Cauchy-Riemann equations based on the graphs (contour map of level curves) of the real and imaginary parts of w = f(z) = z^2 (the level curves are orthogonal (perpendicular) to each other -- they are orthogonal trajectories). The gradient vectors of u and v will be orthogonal when they are nonzero. Discussion of the necessity of the Cauchy Riemann equations as well. (36:36) Example 2: w = f(z) = z^5 + 4z^4 - 5z^3 + 3z - 3. (38:51) Example 3: the exponential function w = f(z) = e^(z) = exp(z) and also check the Cauchy Riemann equations symbolically. (41:58) Example 4: the sine function w = f(z) = sin(z). Its real and imaginary parts can be written in terms of hyperbolic functions. (43:39) View the Cauchy-Riemann equations geometrically in terms of local linearity for Example 4. (47:13) Other things to know for Exam 1.

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