Lecture 1

Prelude

The following documents are a derivative work of my original handwritten notes for MATH 4220 at Cornell University, taken during the Fall semester of 2020. The course was taught by Professor Christian Noack.

The notes are my own interpretation of the lecture material. I don't make any claims regarding their accuracy. Any errors are my own, and do not reflect a mistake by the Professor.

If you find an error somewhere in these notes, please contact me with information about the error; I will add errata.

Topics

The contents of the course can be roughly considered to fall into five main topics.

1. Complex numbers

Generally, this includes all numbers of the form \(a+bi\), where \(a,b\in\mathbb{R}\) and \(i=\sqrt{-1}\).

2. Analytic functions

These form the complex analog to differentiable functions. For example: \[\begin{align}f(z)&=z^2+2iz-7,z\in\mathbb{C}\\f'(z)&=2z+2i\end{align}\] In complex analysis, we will find \(f'\) has a very specific meaning.

3. Complex integration

4. Taylor & Laurent series

Many nice functions can be expressed as an infinite polynomial in \(z\) & \(z^{-1}\). \[ \begin{align} f(z) &= \sum_{n=-\infty}^\infty a_nz^n\\ f(z) &= \sum_{n=0}^\infty b_nz^n \end{align} \]

5. Laplace & Fourier transforms

For example, the Laplace transform of some function \(f(z)\) will be found to be \[g(t)=\int_0^\infty\mathrm{d}z~e^{itz}f(z)\]