Professor Caner Koca
October 9, 2015
Bailey Hall 207
Refreshments will be served in Bailey Hall 204 4:45pm
About a century after the invention of Calculus by Newton and Leibniz, mathematicians such as Euler, Gauss, Riemann and Cauchy discovered and developed a complex-number version of all the key ideas in calculus, such as differentiation and integration. In this new theory, one looks at complex-differentiable functions from complex numbers to complex numbers, and study their properties. This analogy, though very formal, can sometimes lead to really unexpected, surprising and slightly disturbing facts! For example, the complex sine and cosine functions turn out to be unbounded, or the complex exponential function is periodic. In this talk, we will see the basics of the theory of complex functions, and outline some of the similarities and differences between real and complex calculus. I will give special emphasis on how one can visualize and graph some of these complex functions, especially of the so-called multi-valued complex functions, which in turn give rise to some of the fascinating examples of Riemann surfaces.
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