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The zero eigenvalue of the Sturm-Liouville operator (Theorem 33.5). Section 33.9 (This section is one of the most significant for the rest of the course. Work it out with pen and paper): How to solve the eigenvalue problem for a Sturm-Liouville operator in an interval. Example 33.2 (Work it out with pen and paper. Represent graphically solutions to the transcendent equation for eigenvalues (as.
Cauchy’s integral theorem and its use; change of contours. Here we mention that this theorem can be proved without the C1 condition, which guarantees the validity of Green’s formula. Cauchy’s integral formula. Morera’s theorem. Weierstrass’ theorem on convergence of holomorphic functions. Taylor expansion. Identity theorem. Liouville.
Lecture 4: The Cauchy-Riemann equations, harmonic functions, and conjugate harmonic functions (if they exist). Limits, sequences, and series (a review of concepts from real analysis). Same as last class, plus Chapter 2 section 2.1-2.2 (p. 33-35) Jan. 24: Lecture 5: Series and power series, convergence and absolute convergence. Abel's theorem on.
COURSE DESCRIPTION. The course will be an introduction to complex analysis. Topics to be covered include: Holomorphic functions and the Cauchy-Riemann equations, Cauchy's theorem and Cauchy's integral formula, Taylor expansions, entire functions and Liouville's theorem, zeros of holomorphic functions, isolated singularities and Laurent expansions, meromorphic functions, the Residue Theorem.
The following topics will be covered: Holomorphic functions and the Cauchy-Riemann equations, Cauchy's theorem and Cauchy's integral formula, Taylor expansions, entire functions and Liouville's theorem, zeros of holomorphic functions, isolated singularities and Laurent expansions, meromorphic functions, the Residue Theorem, the Maximum Modulus Principle, conformal mappings and linear.
Derivative of a complex function, Cauchy-Riemann equations and holomorphic functions, inverse mapping theorem and the Jacobian, harmonic functions and conjugates, conformal maps and fractional linear transformations. Computer experiments. Complex integration (4 weeks) Multi-calculus review, line integrals and Green’s Theorem. Fundamental.
Theorem, the Maximum Modulus Principle, Liouville’s Theorem, harmonic functions, Taylor and Laurent series, singularities, the Residue Theorem, conformal mapping, normal families, the Riemann Mapping Theorem. Course Objectives The main objective of Complex Analysis is to study the development of functions of one complex variable. Students.