This course is about the special properties associated with differentiation and integration of functions of a complex variable, with some applications to real integration. The techniques have numerous applications in both pure and applied mathematics.
Problem sheet 1
Problem sheet 4
Problem sheet 7
Problem sheet 2
Problem sheet 5
Problem sheet 8
Problem sheet 3
Problem sheet 6
Problem sheet 9 (two-sided)
Solutions to sheet 1
Other solution sheets available upon request!
Section 6.4: The exponential series
None matches the course exactly, but the approach of the first may be closest.
The complex numbers and their basic properties. Exponential, trigonometric, and logarithmic functions. Elements of topology of the complex plane: sequences of complex numbers; open, closed, and compact sets; continuity of complex functions; the extremal value theorem.
Differentiability of functions of a complex variable, the Cauchy-Riemann equations. Power series and functions defined by power series. Contour integration. Existence of the antiderivative.
Cauchy's Theorem. Computation of real integrals. Local theory for analytic functions; uniform convergence of power series, Weierstrass?s M-test. Cauchy's Integral Formula, Taylor's Theorem, Cauchy Estimates. Liouville's Theorem. The Fundamental Theorem of Algebra. Zeros of analytic functions. Topological properties of the zeros.
Functions analytic in an annulus, Laurent's expansion. Classification of singularities. The Residue Theorem and applications to real integrals.