Math 431, Complex Analysis, Spring 2021
Final: Monday May 24th 4:40-6:20pm
Section: 40216
Official departmental syllabus
For homework, you will need to make pdfs of your written work. You may use a scanner, or your phone:
If the resulting file size is too big, try one of the following:
Useful links:
List of calculus formulas and theorems
These are the notes I make for class, they are probably not of much use to anyone else.
Random screenshots.
Found on the internet:
Here's some motivation for why we have definitions and rigorous arguments:
To a first approximation the method of science is “find an explanation and test it thoroughly”, while modern core mathematics is “find an explanation without rule violations”. The criteria for validity are radically different: science depends on comparison with external reality, while mathematics is internal.
The conventional wisdom is that mathematics has always depended on error-free logical argument, but this is not completely true. It is quite easy to make mistakes with infinitesimals, infinite series, continuity, differentiability, and so forth, and even possible to get erroneous conclusions about triangles in Euclidean geometry. When intuitive formulations are used, there are no reliable rule-based ways to see these are wrong, so in practice ambiguity and mistakes used to be resolved with external criteria, including testing against accepted conclusions, feedback from authorities, and comparison with physical reality. In other words, before the transition mathematics was to some degree scientific.
This is an example of the incomprehensible nonsense that precise definitions replaced:
When you need to think very hard about something, try:
Lying down
Mathematicians usually have a hard time explaining to their partner that the times when they work with most intensity is when they are lying down in the dark on a sofa.
or maybe:
Walks
One very sane exercise, when fighting with a very complicated problem (often involving computations), is to go for a long walk (no paper or pencil) and do the computation in one’s head (irrespective of the first feeling “it is too complicated to be done like that”). Even if one does not succeed it trains the “live memory” and sharpens the skills.