The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...

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Analytic continuation of a function

Let $$f(z) = A_0 + A_1(z-a) + A_2(z-a)^2 + \cdots$$ converge in some disk $|z - a| < r$. Following Weyl, we magically re-arrange this power series at point $b$ in this disk and the power series ...