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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;
- Quickly cover classification of zeros and poles of analytic functions
along with giving effective methods for computing residues of isolated
singularities.
- Cover complex analytic techniques to compute a wide variety of definite
real variable integrals which are not accessible by other means.
- Use the above techniques to find inverse Laplace transforms.
- Cover the argument principle and Rouche's theorem with applications to
control theory via Laplace transforms.
- 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.
- Introduce the Poisson kernel for the disk and use it as a mechanism to
cover Fourier series and the Fourier transform.
- 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.
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