From Wikipedia, the free encyclopedia
In mathematics , there are two types of Euler integral :[1]
1. The Euler integral of the first kind is the beta function
B
(
x
,
y
)
=
∫
0
1
t
x
−
1
(
1
−
t
)
y
−
1
d
t
=
Γ
(
x
)
Γ
(
y
)
Γ
(
x
+
y
)
{\displaystyle \mathrm {\mathrm {B} } (x,y)=\int _{0}^{1}t^{x-1}(1-t)^{y-1}\,dt={\frac {\Gamma (x)\Gamma (y)}{\Gamma (x+y)}}}
2. The Euler integral of the second kind is the gamma function
Γ
(
z
)
=
∫
0
∞
t
z
−
1
e
−
t
d
t
{\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}\,\mathrm {e} ^{-t}\,dt}
For positive integers m and n , the two integrals can be expressed in terms of factorials and binomial coefficients :
B
(
n
,
m
)
=
(
n
−
1
)
!
(
m
−
1
)
!
(
n
+
m
−
1
)
!
=
n
+
m
n
m
(
n
+
m
n
)
=
(
1
n
+
1
m
)
1
(
n
+
m
n
)
{\displaystyle \mathrm {B} (n,m)={\frac {(n-1)!(m-1)!}{(n+m-1)!}}={\frac {n+m}{nm{\binom {n+m}{n}}}}=\left({\frac {1}{n}}+{\frac {1}{m}}\right){\frac {1}{\binom {n+m}{n}}}}
Γ
(
n
)
=
(
n
−
1
)
!
{\displaystyle \Gamma (n)=(n-1)!}
See also
References
^ Jeffrey, Alan; and Dai, Hui-Hui (2008). Handbook of Mathematical Formulas 4th Ed. Academic Press. ISBN 978-0-12-374288-9 . pp. 234–235
External links and references