By expanding the beta function
, Equation (10) can be further simplified:
In this work, we continue that line of investigation considering the incomplete beta function
The Beta function
B(a,b) of two variables is defined by
where B(z; s, w) is the incomplete beta function
which is equal to
Then the beta function
for the coupling constant is derived.
No doubt the classical beta function
B([alpha], [beta]) is one of the most fundamental special functions, because of its precious role in several field of sciences such as mathematical, physical, and statistical sciences and engineering.
[??](z) is closed using a presumed shape with a Beta function
as follows :
Using beta function
for (13), the inner integral reduces to
with the best possible constant factor B([lambda]/2, [lambda]/2) ([lambda] > 0, B(u, v) is the beta function
) (see ).
To the observed data, a version of the beta function
used by WANG & ENGEL (1998), called WE model, with f(T) ranging from 0 to 1, was fitted.
B ([alpha], [beta]) denotes the beta function
and the parameters above are all positive real numbers.
One year later, Euler introduced the beta function
defined for a pair of complex numbers a and b with positive real parts, through the integral