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: • Why 0! = 1
The Gaussian Integral: • The Gaussian Integral
Timestamps
Motivation for the Gamma Function: • The Gamma Function
Evaluating the Gamma Function with Integration By Parts: • The Gamma Function
Properties of the Gamma Function: • The Gamma Function
Gamma Function of 1/2: • The Gamma Function
Gamma Function of Positive Half-Integers: • The Gamma Function
Looking for a specific problem or topic? Try checking my website:
www.blacktshirtmathprofessor....
► 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: creativecommons.org/licenses​