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The Omega Constant and Lambert W Function

A personal curiosity led to learning about the "omega constant" (0.56714329...) and the related Lambert W function (named for Johann Lambert). W(z) is a function such that W(z) * e ^ (W(z)) = z for any complex number z. The omega constant Ω = W(1) and is the point where the function e^x crosses the function 1/x, or the fixed point of the function "e to the minus x". Both the function and constant are deceptively simple, but cannot be described using only arithmetic operators and elementary functions. We explore different ways to approximate them computationally, and their relationship to fixed points, a topic we have discussed at length in earlier videos. 0:00 Introduction 0:46 The omega constant 1:09 Computing the omega constant as a fixed point (Haskell) 1:59 The Lambert W function 3:03 Approximating the Lambert W function (Python) 3:36 The relationship between Lambert W and exponential fixed points 4:34 Graphing W(z) on the complex plain You can see our video and fixed points at    • Fixed Point Iteration: Examples, Anal...   To explore the Lambert W function in more depth, we recommend starting with: https://en.wikipedia.org/wiki/Lambert... https://mathworld.wolfram.com/Lambert... Note: The Lambert W function is multibranched, in that it has multiple possible solutions for a particular value. We focus exclusively on the main branch in this video. Background music features Cherry Audio Sines and GX-80, and Arturia CMI V and Tape Mello. _______________ Please subscribe to this channel for cultural content coming out regularly, and please consider supporting us: Merch Store: https://shop.catsynth.com Ko-Fi: https://ko-fi.com/catsynth Patreon:   / catsynth