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Here is a partial list of topics covered this quarter:
Finding residues and using the residue theorem
 | Classifying Singular Points and more residue calculations |
| Residue Calculations and Residue Theorem |
| Residues and Poles |
| Classification of Zeros and Poles |
| Integrating ratios of polynomials |
| Fourier Integrals |
| Integrating ratios of trignometric polynomials |
| Integration involving Branch cuts
(will not be on the final.) |
Applications of the residue theorem
| Argument Principal and Rouche's theorem |
| Open mapping principle. |
| Inverse Laplace Transforms |
Properties of analytic and harmonic functions
| Principle of analytic continuation. |
| Open mapping principle for analytic functions. |
| Maximum Modulus Principle for analytic functions |
| The Maximum and Minimum principle for harmonic functions. |
| Uniqueness of the Dirichlet problem for harmonic functions
on bounded domains. |
Conformal Transformations
| Conformal Mapping Properties of basic analytic functions
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| Mappings involving 1/z |
| Fractional Linear Transformations |
| Conformal Maps and Inverse Functions |
| Mapping Properties of functions involving e^z. |
| Mapping properties of z --> z^2 . |
Harmonic functions, the Dirichlet problem and simple fluid flow
| Finding Harmonic Conjugates (knowing they exist on simply
connected domains.) |
| Laplacian under conformal change of variables and in
particular know that conformal change of variables preserves
harmonic functions. |
| Steady state temperature problems. |
| Electrostatic potentials in 2-dimensions |
| 2D-static irrotational & incompressible fluid flow |
| How to solve the Dirichlet problem on the disk with polynomial boundary data. |
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