The Gamma function is an amazing integral:
Gamma(x) = INTEGRALt=0 to infinity tx-1e-t dt .
Using integration by parts, you can show that this function satisfies the property
Gamma(x) = (x-1) Gamma(x-1).
Using Gamma(1)=1, you can calculate Gamma(2), Gamma(3),…
Does this remind you of anything?
Surprise: the Gamma function satisfies Gamma(n) = Factorial(n-1).
(I would have used the notation “!” but you might think I was just excited!)
So you can think of the Gamma function as being a continuous form of the factorial function. It satisfies lots of cool properties; here is just one:
Gamma(1/2) = Sqrt[Pi].
Presentation Suggestions:
Calculus students might be challenged to compute Gamma(2), Gamma(3), etc. and discover the connection with the factorial function. You may wish to assign the integration by parts as a homework exercise prior to presenting this Fun Fact.
The Math Behind the Fact:
The Gamma function is an important function in analysis, complex analysis, combinatorics, and probability.
How to Cite this Page:
Su, Francis E., et al. “Gamma Function.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.
Fun Fact suggested by:
Michael Moody