We will start by reviewing how complex numbers behave and then talk about complex differentiability and its consequences, Cauchy-Riemann equations.

We will give some elementary examples of complex differentiable functions, aka holomorphic functions. We may also briefly talk about Mobius transformations which are the only complex differentiable functions from the Riemann sphere, aka the complex projective line, onto the Riemann sphere.

I am not promising but we may find time to talk about the fascinating world of Harmonic Functions; the real and imaginary components of a holomorphic function are actually harmonic!

The first surprising result in complex analysis is that all holomorphic functions are analytic, i.e. they have their Taylor series converging to them. For this result we need some integration theory and the Cauchy Integral Formula. The Laurent series and the residue theory then follow immediately.

The next interesting result that we want to cover is the Weierstrass Factorization Theorem for which the key step is the convergence of a sequence of holomorphic functions also to a holomorphic function. There are several ways to establish this and Conway does it using metrics. So we will talk about metrics at some point.

The course will conclude by studying three classical and fascinating functions; Sine, Gamma and the Riemann Zeta functions.

Along the way we will talk about some spectacular results such as the Morera's theorem, Goursat's theorem, Riemann Mapping theorem etc!

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