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diff --git a/ClassJuly5.page b/ClassJuly5.page index 2e9314d..a40482a 100644 --- a/ClassJuly5.page +++ b/ClassJuly5.page @@ -4,10 +4,10 @@ The goal of the lecture (and the only take-away point, as far as the rest of the course is concerned!) was to prove the following two theorems: -Theorem 1: If $f:A \to \C$ is a holomorphic function on an annulus, then it has a Laurent series expansion +Theorem 1: If $f:A \to \mathbb{C}$ is a holomorphic function on an annulus, then it has a Laurent series expansion $$ f(z) = \sum_{n = -\infty}^{\infty} c_n z^n = \cdots \frac{c_{-2}}{z^2} + \frac{c_{-1}}{z} + c_0 + c_1 z + c_2 z^2 + \cdots $$ -Theorem 2: If $f: D \to \C$ is a holomorphic function on a disk, then it has a power series expansion +Theorem 2: If $f: D \to \mathbb{C}$ is a holomorphic function on a disk, then it has a power series expansion $$ f(z) = \sum_{n = 0}^{\infty} c_n z^n = c_0 + c_1 z + c_2 z^2 + \cdots $$ The reason we discussed convergence of Fourier series was to give some taste for the type of mathematical analysis that goes in to proving things rigorously using Fourier series. |