120B Applied Complex Analysis
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Course Goals

120B Catalog Description:  Applications of the residue theorem. Conformal mapping and applications to potential theory, flows, and temperature distributions. Fourier transformations. Laplace transformations, and applications to integral and differential equations. Selected topics such as Poisson’s formula, Dirichlet’s problem, Neumann’s problem, or special functions. 

Specifics to Spring 2018 version of 120B. We will cover Chapters 6-9, and selected parts of chapters 10-12 of Brown and Churchill, "Complex Variables and Applications," 9th edition. Topics will include;

  1. Quickly cover classification of zeros and poles of analytic functions along with giving effective methods for computing residues of isolated singularities.
  2. Cover complex analytic techniques to compute a wide variety of definite real variable integrals which are not accessible by other means.
  3. Use the above techniques to find inverse Laplace transforms.
  4. Cover the argument principle and Rouche's theorem with applications to control theory via Laplace transforms.
  5. Discuss conformal mappings, mostly by numerous examples. Show how these mappings are a useful aid in solving partial differential equations involving the Laplacian in two dimensions.
  6. Introduce the Poisson kernel for the disk and use it as a mechanism to cover Fourier series and the Fourier transform.
  7. Other topics will be covered as time permits. Hopefully we will at least have time to discuss the theory behind series solutions to ordinary differential equations with analytic coefficients.