Here's another cool identity from Euler. You can start by defining the factorial of a number, denoted , as that number multiplied by all the numbers below it. Mathematically, this means . We can also write this recursively, as
Taking , the first few values indexed by are . Taking the factorial as a function, it seems that it can grow far more quickly than any exponential function. This can be easily checked by using the taylor series of :
This sets a lower bound on the growth of . An obvious upper bound is , so we can roughly say that the factorial grows like as approaches . However, doesn't fully satisfy the functional equation , so there must be lower-order terms. These can be found with Stirling's approximation.
Here's a challenge: can we define the factorial for all real numbers? How about complex ones as well? It turns out we can, and this is where the gamma function can be introduced:
This seemingly comes out of nowhere, but it relies on a following neat observation:
Thus it satisfies the same functional formula, . Evaluating the full integral gives whenever is an integer. We now have a factorial function, the Gamma function, which is continuously defined.
Due to the functional equation, this function blows up at all negative values. However, it stays finite for non-integer negative values. Thanks to the Weierstrass factorization theorem, we can factorize this function (or it's reciprocal) as a product of its poles (or zeroes) as:
Where . This will take a bit of work to justify. We can start with the peculiar identity
To prove this, we can expand as and introduce a strange factor to get
The leftmost fraction expands to
Which approaches as . This must mean that
It takes some more work to prove that this works with any argument (it's definitely a challenging exercise). I'll skip the details here though. From here we can find the Weierstrass factorization
This leads to a really interesting functional equation for the gamma function, known as Euler's reflection formula. The Weierstrass factorization of is given by:
The big terms suggest that we can relate this with in some way. We can derive:
Where . Replacing the separated terms in the Weierstrass factorization of yields:
Appealing to the functional equation leads us to simplify as , giving us the full reflection identity: