Ever wondered what (1/2)! might be, or thought about finding smooth curves passing through factorial points? Well this can be done using the Gamma function, one of the most popular extensions of the factorial. In this video I outline the derivation of its main integral formula, consider some particular values and relationships, as well as outline other important representations and identities.
For proofs of some of the formulas mentioned here, I have a separate video: https://youtu.be/rN0Ap0Ba7R4
Timecodes:
0:00 - Main Forms and the Functional Equation
10:29 - Smaller Formulas
13:34 - Particular values, Γ(1/2 ± n), and Approximations
17:49 - Reflection Formula and Products/Ratios
21:34 - Duplication Formula and Multiplication Theorem Mention
24:20 - Beta Function and Derivative Possibilities
Sources and other tidbits:
Gamma Function Outline: https://en.wikipedia.org/wiki/Gamma_function
More on the Beta Function: https://en.wikipedia.org/wiki/Beta_function
Gamma Function as Area Under a Curve in Desmos: https://www.desmos.com/calculator/e4zsbt5bsn
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