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© 1999-2010 by Francis Edward Su
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From the Fun Fact files, here is a Fun Fact at the Medium level:

Gamma Function

The Gamma function is an amazing integral:

Gamma(x) = INTEGRALt=0 to infinity tx-1 e-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. <http://www.math.hmc.edu/funfacts>.

Keywords:    calculus, continuous factorial
Subjects:    calculus, analysis
Level:    Medium
Fun Fact suggested by:   Michael Moody
Suggestions? Use this form.
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