The Gamma function is an amazing integral:
Gamma(x) = INTEGRAL_{t=0 to infinity}
t^{x1} e^{t} dt .
Using integration by parts, you can show that this
function satisfies the property
Gamma(x) = (x1) Gamma(x1).
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(n1).
(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.
<http://www.math.hmc.edu/funfacts>.
