Factorial, Gamma Function, and More
🏛 A Project Cauchy Op-ed
Named after French mathematician Augustin-Louis Cauchy, Project Cauchy column is where I invite some of the HFI Programming club members to provide neat proofs or explanations about some number theory puzzles. I highly suggest that you read these articles with a pencil and paper so you can sketch things out and scribble solutions to exercises as you come across them. This week, Travor is demonstrateing us one of the most commonly used extension of the factorial function to complex numbers.
It is well-known that factorial is defined by the following recursive relation,
with , but however it is possible to generalize this operation to complex numbers. Let's begin our journey of generalizations!
First generalization: integral representation
It can be easily shown that the following integral satisfies
Differentiation with respect to on both side for times gives
Setting gives us the first generalization of factorial: the Gamma function.
In fact, by integration by parts we can show that the Gamma function satisfies the recursive relationship:
Furthermore, we have the relation
Digression: the Beta function
To convenience our derivation process, let's define the Euler integral of the first kind: the Beta function :
which can be seen as a convolution between two power functions:
If we were to perform Laplace transform on both side, we get
If we juxtapose this fact with the relation
then we see that
which will be extremely useful for us to generalize factorial even further.
evaluated at complex numbers
Via taking absolute variable, we know that the Gamma integral converges whenever
which means that this improper integral is converges absolutely on the right half plane. Hence, a stronger definition is needed for us to expand it to the entire complex plane.
as a limit
Let's consider a sequence of functions:
Then by the exponential limit we see that
in a pointwise sense. However, it is possible to show that this sequence converges uniformly for . Let's set such that for all we have
Then, let's consider the interval :
where we define as
By the uniform continuity of on , all we need is to prove that converges uniformly to . In fact, for all we can use the Taylor expansion of logarithm to obtain
As a result, we conclude that converges uniformly to , which allows us to interchange the limit operation and integral to obtain
In the following procedure, we are going to expand the left hand side limit in a subtle sense so that the right hand side can be analytically continued to the left half plane.
Expanding the integral sequence
Performing a change of variables on (3) gives
In fact, the right hand side can be expressed by Beta function as in (3), which yields
Continuous application of (1) gives us
which transforms Gamma function into a product representation, however this expression still looks ugly, why not go deeper?
Weierstrass product for
First, let's turn this equation up side down to obtain
Then, employing the fact that we can replace the logarithm with harmonic numbers:
Now, if we take logarithm on the product, we have
As a result, the product converges absolutely for all , giving us the Weierstrass product representation of Gamma function:
which allows us to analytically continue to the entire complex planes except for nonpositive integers:
Remark: (4) also reveals is non-zero for all
Now, we successfully expanded factorial to the entire complex plane except at negative integers, but Gamma function has some other brilliant properties. Let's have a look at some of them:
Alternative Definition for Euler-Mascheroni constant
From the last article, we know that Euler-Mascheroni constant is defined by
However, it is possible for us to create a new definition for by using .
Remark: I hypothesize this explains why they name the function and the constant since they are highly correlated.
To begin with, we take logarithm on (4) to get
If we were to define Digamma function as the logarithmic derivative of , then
If we were to move the term into the summation, we deduce the standard definition for Digamma function.
Plugging gives . Because , we also know that . Combining this with the original integral definition for gives this elegant integral identity to represent Euler-Mascheroni constant:
Integral representation for Digamma function
While deriving (5) in the previous section, we introduce the Digamma function, so why don't we do some calculus on that
Well, the constant lying outside is not beautiful, so why not use (6):
Combining all these gives us the ultimate integral definition for :
In this blog, we first use the technique of differentiation under integral to deduce an integral representation for factorial that introduces the concept of Gamma function . Then, by using an identity connecting and , we obtain a product formula that turns into a meromorphic function on . Using this newly obtained product formula, we are also able to discover some new identities. In fact, Gamma function is a function that often appears in the field of analytic number theory, and we will begin future investigation based on the following identity (which you could try prove it yourself):