derivatives of the functions in equation (12.76) with respect to say xj, 1 ≤j≤n, one
finds
∂F 1
∂x j+
∂F 1
∂y 1
∂y 1
∂x j+···+
∂F 1
∂ym
∂y m
∂x j =0
∂F 2
∂x j+
∂F 2
∂y 1
∂y 1
∂x j+···+
∂F 2
∂ym
∂y m
∂x j =0
..
.
..
.
..
.
∂F n
∂x j +
∂F n
∂y 1
∂y 1
∂x j+···+
∂F n
∂ym
∂y m
∂x j =0
(12 .78)
The system of equations (12.78) can be solved by Cramers rule and because the
Jacobian determinant is different from zero the system of equations (12.78) has a
unique solution for the various first partial derivatives.
Higher derivatives can be obtained by differentiating the first order partial
derivatives.
The Gamma Function
One definition of the Gamma function is given by the integral
Γ(x) =
∫∞
0
ξx−^1 e−ξdξ (12 .79)
The value x= 1 substituted into the equation (12.79) produces the result
Γ(1) =
∫∞
0
e−ξdξ =−e−ξ
]∞
0 = 1 (12 .80)
so that Γ(1) = 1. Substitute x=n+ 1 , a positive integer, into equation (12.79) and
then integrate by parts to obtain
Γ(n+ 1) =
∫∞
0
e−ξξndξ =
[
−ξne−ξ
]∞
0 +
∫∞
0
nξ n−^1 e−ξdξ
which simplifies to the recurrence relation
Γ(n+ 1) = nΓ(n), n > 0 , an integer (12 .81)
In general, for any real positive value for xwhich is less than unity, one can show
that Γ(x)is a particular solution of the functional equation
Γ(x+ 1) = xΓ(x), x > 0 (12 .82)
