The factorial of 3.5: the gamma function, derived from…
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The factorial of 3.5: the gamma function, derived from binomial coefficients
The factorial of an integer can, perhaps surprisingly, be evaluated even at non-integer entries. For example, you might even see these factorials of non-integers appearing in formulas for the volumes of high dimensional balls (usually, these factorials appear in the guise of the 'gamma function').
The extension of the factorials to non-integers is usually done with a certain integral formula, but Euler's originally derivation actually used some simple combinatorial identities, which he realized allowed him to write down a formula for x! which only involved factorials of integers and certain standard arithmetic operations. This let Euler define x! in general, as a certain limit.
At https://hidden-phenomena.com/articles/gamma , you can see this derivation in full -- it's quite cool!
#science
The factorial of an integer can, perhaps surprisingly, be evaluated even at non-integer entries. For example, you might even see these factorials of non-integers appearing in formulas for the volumes of high dimensional balls (usually, these factorials appear in the guise of the 'gamma function').
The extension of the factorials to non-integers is usually done with a certain integral formula, but Euler's originally derivation actually used some simple combinatorial identities, which he realized allowed him to write down a formula for x! which only involved factorials of integers and certain standard arithmetic operations. This let Euler define x! in general, as a certain limit.
At https://hidden-phenomena.com/articles/gamma , you can see this derivation in full -- it's quite cool!
#science
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