Cauchy's Residue Theorem: An In-Depth Introduction, Derivation, and Explanation with Applications

Hey everyone! Today, I wanted to take this opportunity to give a foundational explanation of Cauchy's Residue Theorem! Sorry that the format is kinda wack. When I uploaded this, I was in high school, and this was my first YouTube video :) My simple definition of Complex Analyticity comes from MIT's OCW and is in reference to complex-valued functions specifically. There are some other definitions, but they basically all imply the same things:) https://ocw.mit.edu/courses/18-04-complex-... Foundations - 00:00 00:00 - Intro 01:00 - Analytic/Holomorphic Functions and Singularities of Complex Functions 05:20 - Laurent Series and its Integration+Residues at Singularities 09:42 - Limit Residue Formula+Derivation 14:40 - Applying Stokes’ Theorem and Cauchy’s Residue+Integral Theorems 17:35 - Jordan’s Lemma “Proof” Applications - 21:45 21:45 - Applications Intro 22:52 - Real Integral Examples (Contour Integration) 22:52 - #1 (Arctangent) 28:19 - #2 (Using Euler’s Formula) 32:38 - #3 (Unit Circle) 37:48 - #4 (The Dirichlet Integral) 44:51 - Inverse Laplace Transform Intro 47:47- Inverse Laplace Transform Examples (Differential Equations) 47:47 - #1 (Residues of Poles of Higher Orders) 51:50 - #2 (Binomial Theorem?!) 55:41 - Outro :) Other Resources: Examples and Practice Problems: https://madasmaths.com/archive/maths_bookl... ‪@qncubed3‬ 's Jordan's Lemma Proof (His channel is also a great resource for complex analysis in general!)    • Complex Analysis: Jordan's Lemma   Inverse Laplace Transform/Bromwich Contour In-Depth:    • The Bromwich Integral and The Inverse Lapl...