ABSTRACT
The beta function, or Eulerian integral of the rst kind, is dened by the
Euler integral ([9], [56], [284])
B(x; y) :=
Z
t
x1
(1 t)
y1
dt
Re(x) > 0; Re(y) > 0
; (5.1)
and is related to the gamma function through
B(x; y) :=
(x)(y)
(x+ y)
: (5.2)
The representation (5.1) applies only when both arguments have positive real
parts, but (5.2) extends the denition to any pair of arguments where the
ratio is dened. Similarly, the innite product expression
B(x; y) =
(x+ y)
xy
Y
n=1
1 +
x+y
n
1 +
x
n
1 +
y
n
(5.3)
provides an analytic continuation of the function.
