Class Log

Friday 12/11/15: More examples of the mapping problem. Statement of the Riemann mapping theorem. Brief discussion of the Schwarz--Christoffel formula describing a mapping from the upper half plane to the interior of a rectangle.

Wednesday 12/9/15: The mapping problem: given open sets U and V of the complex plane, find a complex differentiable function g: U --> V which is one-to-one and onto. Given such a function g, the derivative g' is nowhere zero on U, so that (1) g preserves angles and (2) the inverse of g is also complex differentiable. Examples using elementary functions.

Monday 12/7/15: Another example: using an LFT to map the region bounded by two circles meeting at right angles to the positive quadrant. Proofs of properties of LFT's stated last time.

Friday 12/4/15: Linear fractional transformations f(z)=(az+b)/(cz+d). LFT's are invertible functions from the extended complex plane (the complex numbers together with the point ∞) to itself. If C is a circle or line then f(C) is a circle or line. If A,B,C are 3 distinct complex numbers, then the LFT f(z)=((Z-A)/(Z-B))/((C-A)/(C-B)) maps A,B, and C to 0, ∞, and 1. Example: The LFT f(z)=-i(z-1)/(z+1) maps the disc with center the origin and radius 1 to the upper half plane.

Wednesday 12/2/15: Application to fluid flow. For an incompressible irrotational 2-dimensional flow the velocity field is given by grad φ for some harmonic function φ. If f= φ + i ψ is complex differentiable, then the level curves of ψ give the streamlines of the flow (the paths of particles in the fluid). Solution of fluid flow problems by using a complex differentiable function to map the original domain to a simpler domain.

Monday 11/30/15: Harmonic functions. If f=u+iv is complex differentiable then u and v are harmonic. Conversely if u is harmonic on a simply connected domain U then u is the real part of a complex differentiable function on U.

Friday 11/27/15: No class (Thanksgiving).

Wednesday 11/25/15: More examples of computing real integrals via the residue theorem.

Monday 11/23/15: Computing real integrals using the residue theorem: rational functions and trigonometric integrals.

Friday 11/20/15: Computing the residue at a pole.

Wednesday 11/18/15: Riemann's removable singularity theorem. The Casorati--Weierstrass theorem. Cauchy's residue theorem.

Monday 11/16/15: Proof of existence of Laurent series expansions for complex differentiable functions near an isolated singularity. Integral formula for the coefficients of the Laurent series. The residue of a complex differentiable function at an isolated singularity.

Friday 11/13/15: Laurent series (continued). Removable singularities, poles, and essential singularities.

Wednesday 11/11/15: No class (Veteran's day)

Monday 11/9/15: Defining functions via power series. Radius of convergence of a power series. Computing the radius of convergence using the ratio test. Laurent series.

Friday 11/6/15: Example of a real function which has derivatives of all orders but does not have a power series expansion. Theorem: If f is a complex differentiable function then it has a power series expansion at every point of its domain, and the expansion is valid in the largest open disc which is contained in the domain.

Wednesday 11/4/15: Review of real power series expansions from Math 132. The geometric series 1/(1-x)= 1+x+x2+x3+..., valid for |x|<1.

Monday 11/2/15: Proof of the fundamental theorem of algebra using Liouville's theorem. Proof of the generalized Cauchy integral formula using differentiation under the integral sign.

Friday 10/30/15: Generalized Cauchy integral formula (Cauchy integral formula for derivatives). Consequences: f complex differentiable implies f has complex derivatives of all orders. Liouville's theorem: Suppose f is a complex differentiable function with domain the whole complex plane. If f is bounded then f is constant.

Wednesday 10/28/15: More examples of critical points of u and v, for f=u+iv complex differentiable. The monkey saddle. Proof of Cauchy's integral formula.

Monday 10/26/15: Cauchy's integral formula. Special case: Gauss' mean value theorem. Application: if f=u+iv is complex differentiable, then all critical points of u and v are saddle points (cannot be local max or local min). Examples.

Friday 10/23/15: The integral of 1/(z-a) around a simple closed curve oriented counterclockwise equals 2πi if a is inside C and 0 if a is outside C. Application: computing real integrals of rational functions.

Wednesday 10/21/15: A complex differentiable function on a domain with no holes has an antiderivative (constructed via contour integration).

Monday 10/19/15: Constructing antiderivatives via contour integration. Cauchy's theorem (aka the Cauchy-Goursat theorem).

Friday 10/16/15: Bounding contour integrals in terms of the length of the curve C and the maximum of |f(z)| for z in C. The triangle inequality. Second part of the complex fundamental theorem of calculus (preliminary discussion).

Wednesday 10/14/15: Changing the orientation of the contour reverses the sign of the integral. Computing contour integrals using antiderivaties (first part of complex fundamental theorem of calculus).

Tuesday 10/13/15: Contour integration. Definition and examples. Integral is independent of choice of parametrization of the curve.

Friday 10/9/15: Computing complex derivatives: Linearity, product rule, quotient rule, chain rule, derivatives of basic functions (powers of z, exponential function, sine and cosine, logarithm).

Wednesday 10/7/15: The Cauchy--Riemann equations. Conformal property (complex differentiable functions preserve angles between curves at points where the derivative is nonzero).

Monday 10/5/15: Complex differentiability.

Friday 10/2/15: Review of differentiability of real functions of two variables.

Wednesday 9/30/15: The function f(z)=1/z corresponds to rotation of the sphere through pi radians about the x-axis under stereographic projection

Monday 9/28/15: Complex rational functions. The function f(z)=1/z. Stereographic projection.

Friday 9/25/15: Geometric interpretation of the complex logarithm. Defining (multivalued) complex powers wz using the logarithm.

Wednesday 9/23/15: The complex logarithm (principal value Log z and multivalued version log z).

Monday 9/21/15: The complex sine and cosine functions.

Friday 9/18/15: The complex exponential function.

Wednesday 9/16/15: Review of the definition of the (real) exponential function f(x)=ex and its properties.

Monday 9/14/15: Complex mappings. Examples: Polynomials. Linear functions f(z)=az+b (where a and b are complex numbers). Power functions f(z)=zn (where n is a positive integer).

Friday 9/11/15: The fundamental theorem of algebra. Motivation from linear algebra. Finding the complex solutions of polynomial equations. Roots of unity.

Wednesday 9/9/15: Complex numbers. Arithmetic operations. Geometric interpretation. Conjugation. The fundamental theorem of algebra.