Is it possible to define (1/2)!? Can we define the factorial of non-integer values? As it turns out, we can! To do this we need to introduce an important function from higher-level mathematics called the gamma function. Don't worry: I keep everything at a level that only requires understanding calculus 2, specifically integration by parts and improper integrals. We go through everything: motivating the gamma function, some of it's basic values, how to relate factorials to the gamma function, and proving an important property of the gamma function. Two prerequisite videos are linked below as well as timestamps.
Why 0! = 1: https://youtu.be/ZSJLaWg94Do
The Gaussian Integral: https://youtu.be/FiCKDl1rKl4
Timestamps
Motivation for the Gamma Function: https://youtu.be/-GXHHg6kFZM?t=8
Evaluating the Gamma Function with Integration By Parts: https://youtu.be/-GXHHg6kFZM?t=358
Properties of the Gamma Function: https://youtu.be/-GXHHg6kFZM?t=825
Gamma Function of 1/2: https://youtu.be/-GXHHg6kFZM?t=1473
Gamma Function of Positive Half-Integers: https://youtu.be/-GXHHg6kFZM?t=1751
Looking for a specific problem or topic? Try checking my website:
https://www.blacktshirtmathprofessor.com/videos
► Artist Attribution
Music By: "After The Fall"
Track Name: "Pieces"
Music Published by: Chill Out Records LLC
License: Creative Commons Attribution 4.0 International (CC BY - 4.0)
Full license here: https://creativecommons.org/licenses