The gamma function is defined as the integral of
t^(a-1) exp(-t) integrated from 0 to infinity.
The gamma function is the generalization of the factorial function.
gamma(1) = factorial(0) = 1
gamma(2) = factorial(1) = 1
gamma(3) = factorial(2) = 2
gamma(4) = factorial(3) = 6
.
.
.
gamma(N) = factorial(N-1)
One variety of incomplete gamma function is defined as the integral of
t^(a-1) exp(-t) integrated from 0 to x.
Another variety of incomplete gamma function is defined as the integral of
t^(a-1) exp(-t) integrated from x to infinity. In fact, this seems to be
the most popular choice. Probably because it is easy to describe
when the parameter "a" has an integer value.
Both varieties are used, and both called the incomplete gamma function.
Also some authors use the name incomplete gamma function to refer to one
of the above varieties divided by the value of the gamma function. That
is, they want the incomplete gamma function to be normalized by the gamma
function.
Which of the four varieties an author uses depends on the application.
The different varieties have simple relationships to each other, so it
doesn't matter much which one you use.
However, this confusion in names requires you to say which integral you
are calculating for the incomplete gamma function.
I shall give distinct names to the 4 varieties of incomplete gamma
function.
x
Define GRL(x,a) = [1/gamma(a)] * integral [ t^(a-1) * exp(-t) ] dt
0
infinity
Define GRU(x,a) = [1/gamma(a)] * integral [ t^(a-1) * exp(-t) ] dt
x
x
Define GL(x,a) = integral [ t^(a-1) * exp(-t) ] dt
0
infinity
Define GU(x,a) = integral [ t^(a-1) * exp(-t) ] dt
x
infinity
Then GL(x,a) = gamma(a) * exp(-x) * sum [ x^(a+j) ]/gamma(a+1+j) ]
j=0
GU(x,a) = gamma(a) - GL(x,a)
GRL(x,a) = GL(x,a)/gamma(a)
GRU(x,a) = GU(x,a)/gamma(a)
Also,
GL(x,a) =
exp(-x) * x^a * [ 1/a + x/[a * (a+1) ] + [x^2] / [a * (a+1)*(a+2)] +...
infinity
= x^a * exp(-x) sum x^j / [a(a+1)...(a+j)]
j = 0
As a test of this formula, show that the value at large x is very close to
gamma(a).
Also, consider the special case where the parameter a is equal to an
integer N.
infinity
Then GL(x,N) = gamma(N) * exp(-x) * sum [ x^(N+j) ]/gamma(N+1+j) ]
j=0
infinity
GL(x,N) = factorial(n-1) * exp(-x) * sum [x^(N+j)]/factorial(N+j) ]
j=0
infinity
GL(x,N) = factorial(n-1) * exp(-x) * sum [x^(N+j)]/factorial(N+j) ]
j=0
infinity
GL(x,N) = factorial(n-1) * exp(-x) * sum ( [x^j]/factorial(j) ])
j=N
N-1
GL(x,N)=factorial(N-1)*exp(-x)*[ exp(x) - sum( [x^j]/factorial[j] )]
j=0
N-1
GL(x,N) = factorial(N-1) [ 1 - exp(-x) * sum( [x^j]/factorial[j] )]
j=0
N-1
GL(x,N) = gamma(N) [ 1 - exp(-x) * sum( [x^j]/factorial[j] )]
j=0
N-1
GU(x,N) = gamma(N) * exp(-x) * sum( [x^j]/factorial[j] )]
j=0
N-1
GRU(x,N) = exp(-x) * sum( [x^j]/factorial[j] )]
j=0
Note that the sum is the first (N+1) terms of the power series for exp(x).
Comments? Questions?
Kermit.