**Description: ** | The complex numbers, described by Leibniz as amphibia between existence and non-existence, are now an important tool for both pure and applied mathematics. They have a fruitful geometric interpretation, provide an algebraic closure to the reals (in the sense that all polynomials with coefficient in C have roots in C), and allow, with a more coherent theory than for real variables, the development of the calculus. The important exponential function, in particular, extends elegantly to the complex domain. This course will concentrate on the differentiation and integration of complex functions and their mapping properties. We will see application of our theory to geometry, dynamics (including the Mandelbrot set), and physics. A working knowledge of elementary calculus is assumed. There will be a weekly problem session attached to the course and regular written assignments. |