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Gamma Function and Beta Function
Definition of the Gamma Function. The Gamma function in a real variable is defined by
Γ (x) =
0
tx−^1 e−tdt
for x > 0 to make sure that the integral converges at t = 0. When x > 1, by integration by parts we get
Γ (x) =
[
−tx−^1 e−t
]t=∞ t=0 + (x^ −^ 1)
0
tx−^2 e−tdt = (x − 1) Γ (x − 1).
From Γ(1) =
0 e
−t (^) = 1 it follows that
Γ (n) = (n − 1)!.
So the Gamma function is the generalization of the factorial function from integer values to real values. The defining formula
Γ (z) =
0
tz−^1 e−tdt
actually defined Γ(z) for z ∈ C with Re z > 0.
Beta Function. A similar analog of the generalization of the binomial coeffi- cient (^) ( m + n m
(m + n)! m! n!
is the Beta function defined by
B (x, y) =
Γ (x + y) Γ(x) Γ(y)
We are going to derive the formula for the Beta function as a definite integral whose integrand depends on the variables x and y. This is done by reversing the order of integration of a double integral. For x > 0 and y > 0 we have
Γ(x)Γ(y) =
0
tx−^1 e−tdt
0
uy−^1 e−udu
Using the transformation u = tv and then the transformation w = t (1 + v), we obtain
Γ(x)Γ(y) =
0
tx−^1 e−tdt
0
tyvy−^1 e−tvdv
0
vy−^1 dv
0
tx+y−^1 e−t(1+v)dv
0
vy−^1 dv
0
wx+y−^1 e−wdw (1 + v)x+y
= Γ (x + y)
0
vy−^1 dv (1 + v)x+y^
from which it follows that
B (x, y) =
0
vy−^1 dv (1 + v)x+y^
Finally we use the transformation v = (^1) −λλ to get the alternative formulation
B (x, y) =
0
λx−^1 (1 − λ)y−^1 dλ,
which is symmetric in x and y.
Relation Between Gamma Function and Sine Function. A very useful case for the Beta function is when x + y = 1 in the above formula, in which case
Γ(x)Γ(1 − x) =
0
vx−^1 dv 1 + v
which by residue calculus applied to the function
zx−^1 dz 1 + z integrated over the contour integral of the boundary of the domain
{ r < |z| < R } − { Re z ≥ 0 , −r ≤ Im z ≤ r } ,
yields π sin πx
Thus we have the following important formula relating the gamma function to the sine function Γ(x)Γ(1 − x) =
π sin πx for 0 < x < 1.
