I post here together two answers of mine on MSE both on the gamma function.
Question by pomme. Definition of the gamma function
“I know that the Gamma function with argument
— in other words
is equal to
. However, the definition of
but how can
be obtained from the definition? WA says it does not converge…”
Your doubt makes sense if for you try using the definition of the gamma function by the integral you have written, because it diverges at
as you stated. I will try to clarify if as follows. This integral representation of the gamma function (I use
instead of
)
holds in the reals if and only if . Using integration by parts we can show that
For we can define
for all negative values of
except
not by the integral
, rather by means of the functional equation
in the form
Then [and]
is convergent. In your example
, so
and
. Hence we obtain
where we use the known value of the integral , which can be evaluated, e.g. from the equality
Similarly, for by
we find
, using
twice. This process is called analytic continuation, but its true understanding requires knowledge of complex analysis.
Question by Amitabh Udayiman. Convergence of this integral
“My statistics text book prescribed by my school states that the integral
is convergent for
.It does not prove the statement. So can anyone please help me prove it? Thanks again!”
My answer. I assume that is a real number. Split the gamma improper integral
into , where
and
1. To prove that the integral is always convergent use the fact that for any real number
the integral
is convergent, by the limit comparison test
with the convergent integral
2. As for consider two cases. (a) If
observe that
, so
is a proper integral. (b) If
, the integrand
behaves like
near
, because
as
. Since
is convergent if and only if , i.e.
, so is
. It follows that
is convergent for