NP-Incompleteness
Path-Independent Line Integrals
13 Apr 2024
This is our second post in the series with my notes on complex integration, corresponding to Chapter 4 in Ahlfors’ Complex Analysis.
In this installment, we’ll cover line integrals that are path-independent, in other words, that depend only on the first and last point of the paths, but not the specific path or curve chosen. We’ll focus on the conditions these integrals have to satisfy for this to hold.
We’ll start with line integrals for the Euclidean plane and then follow with complex line integrals.
Complex Integration
05 Apr 2024
This post inaugurates a series with my notes on complex integration, corresponding to Chapter 4 in Ahlfors’ Complex Analysis.
In this post we’ll cover different definitions of complex integrals - integrals of functions whose domain and image are complex numbers - and some basic properties.
Understanding std::call_once() in C++
01 Mar 2024
In this post, we’ll delve into the function std::call_once() in the C++ STL: why it’s useful, how efficient it is and how it can be implemented. We’ll provide an simple implementation based on locks and a more advanced one based on futexes.
Finally we do a benchmark to compare their performance vs. libraries such as GCC, Clang and Folly.
Bipolar Coordinates and Möbius Transformations
10 Feb 2024
In this post we’ll discuss the bipolar coordinate system and how it can be used to simplify Möbius transformations. It puts together a bunch of concepts we studied in prior posts, so being familiar with the series is helpful.
This is the last post of the series based on holomorphic functions, which correspond to Chapter 3 (Analytic Functions as Mappings) in Ahlfors’ Complex Analysis [1].
Symmetry Points of a Circle
03 Feb 2024
In this post we study the Symmetry Points of a Circle in the context of Complex Analysis. It builds on concepts and results discussed in prior section, so it’s worth getting familiar with the series: