lpetrich
Contributor
I have fond memories from long ago of Abramowitz and Stegun's fat book "Handbook of Mathematical Functions". It is online at Abramowitz and Stegun: Handbook of Mathematical Functions
One of the chapters is "Gamma Function and Related Functions", about an extension of the factorial function. To understand that extension, consider an integral representation of it:
\( n! = \int_0^\infty t^n e^{-t} dt \)
It is easy to find 0! from it, and also to find the factorial recurrence relation, n! = n*(n-1)! from it. One does integration by parts:
\( n! = \int t^n d(-e^{-t}) = - t^n e^{-t} + n \int t^{n-1} e^{-t} dt \)
Can we extend this integral representation to non-integer n? We can indeed do so, and that is the Euler gamma function:
\( n! = \Gamma(n+1) \)
\( \Gamma(z) = \int_0^\infty t^{z-1} e^{-t} dt \)
One can express the exponential function as the limit of a polynomial function:
\( \Gamma(z) = \lim_{n\to\infty} \int_0^n t^{z-1} \left( 1 - \frac{t}{n} \right)^n dt = \lim_{n\to\infty} \frac{n! n^z}{\prod_{k=0}^n (z+k)} \)
This is Euler's formula, and it yields Euler's infinite product:
\( \frac{1}{\Gamma(z)} = z e^{\gamma z} \prod_{n=1}^\infty \left( 1 + \frac{z}{n} \right) e^{-z/n} \)
where γ is the Euler-Mascheroni constant:
\( \gamma = 0.57721\ 56649 \ldots \)
One of the chapters is "Gamma Function and Related Functions", about an extension of the factorial function. To understand that extension, consider an integral representation of it:
\( n! = \int_0^\infty t^n e^{-t} dt \)
It is easy to find 0! from it, and also to find the factorial recurrence relation, n! = n*(n-1)! from it. One does integration by parts:
\( n! = \int t^n d(-e^{-t}) = - t^n e^{-t} + n \int t^{n-1} e^{-t} dt \)
Can we extend this integral representation to non-integer n? We can indeed do so, and that is the Euler gamma function:
\( n! = \Gamma(n+1) \)
\( \Gamma(z) = \int_0^\infty t^{z-1} e^{-t} dt \)
One can express the exponential function as the limit of a polynomial function:
\( \Gamma(z) = \lim_{n\to\infty} \int_0^n t^{z-1} \left( 1 - \frac{t}{n} \right)^n dt = \lim_{n\to\infty} \frac{n! n^z}{\prod_{k=0}^n (z+k)} \)
This is Euler's formula, and it yields Euler's infinite product:
\( \frac{1}{\Gamma(z)} = z e^{\gamma z} \prod_{n=1}^\infty \left( 1 + \frac{z}{n} \right) e^{-z/n} \)
where γ is the Euler-Mascheroni constant:
\( \gamma = 0.57721\ 56649 \ldots \)