Mathematical function
Contour plot of the beta function
In
mathematics , the beta function , also called the
Euler integral of the first kind, is a
special function that is closely related to the
gamma function and to
binomial coefficients . It is defined by the
integral
B
(
z
1
,
z
2
)
=
∫
0
1
t
z
1
−
1
(
1
−
t
)
z
2
−
1
d
t
{\displaystyle \mathrm {B} (z_{1},z_{2})=\int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\,dt}
for
complex number inputs
z
1
,
z
2
{\displaystyle z_{1},z_{2}}
such that
ℜ
(
z
1
)
,
ℜ
(
z
2
)
>
0
{\displaystyle \Re (z_{1}),\Re (z_{2})>0}
.
The beta function was studied by
Leonhard Euler and
Adrien-Marie Legendre and was given its name by
Jacques Binet ; its symbol Β is a
Greek capital
beta .
Properties
The beta function is
symmetric , meaning that
B
(
z
1
,
z
2
)
=
B
(
z
2
,
z
1
)
{\displaystyle \mathrm {B} (z_{1},z_{2})=\mathrm {B} (z_{2},z_{1})}
for all inputs
z
1
{\displaystyle z_{1}}
and
z
2
{\displaystyle z_{2}}
.
[1]
A key property of the beta function is its close relationship to the
gamma function :
[1]
B
(
z
1
,
z
2
)
=
Γ
(
z
1
)
Γ
(
z
2
)
Γ
(
z
1
+
z
2
)
{\displaystyle \mathrm {B} (z_{1},z_{2})={\frac {\Gamma (z_{1})\,\Gamma (z_{2})}{\Gamma (z_{1}+z_{2})}}}
A proof is given below in
§ Relationship to the gamma function .
The beta function is also closely related to
binomial coefficients . When m (or n , by symmetry) is a positive integer, it follows from the definition of the gamma function Γ that
[1]
B
(
m
,
n
)
=
(
m
−
1
)
!
(
n
−
1
)
!
(
m
+
n
−
1
)
!
=
m
+
n
m
n
/
(
m
+
n
m
)
{\displaystyle \mathrm {B} (m,n)={\frac {(m-1)!\,(n-1)!}{(m+n-1)!}}={\frac {m+n}{mn}}{\Bigg /}{\binom {m+n}{m}}}
Relationship to the gamma function
A simple derivation of the relation
B
(
z
1
,
z
2
)
=
Γ
(
z
1
)
Γ
(
z
2
)
Γ
(
z
1
+
z
2
)
{\displaystyle \mathrm {B} (z_{1},z_{2})={\frac {\Gamma (z_{1})\,\Gamma (z_{2})}{\Gamma (z_{1}+z_{2})}}}
can be found in
Emil Artin's book The Gamma Function , page 18–19.
[2]
To derive this relation, write the product of two factorials as
Γ
(
z
1
)
Γ
(
z
2
)
=
∫
u
=
0
∞
e
−
u
u
z
1
−
1
d
u
⋅
∫
v
=
0
∞
e
−
v
v
z
2
−
1
d
v
=
∫
v
=
0
∞
∫
u
=
0
∞
e
−
u
−
v
u
z
1
−
1
v
z
2
−
1
d
u
d
v
.
{\displaystyle {\begin{aligned}\Gamma (z_{1})\Gamma (z_{2})&=\int _{u=0}^{\infty }\ e^{-u}u^{z_{1}-1}\,du\cdot \int _{v=0}^{\infty }\ e^{-v}v^{z_{2}-1}\,dv\\[6pt]&=\int _{v=0}^{\infty }\int _{u=0}^{\infty }\ e^{-u-v}u^{z_{1}-1}v^{z_{2}-1}\,du\,dv.\end{aligned}}}
Changing variables by u = st and v = s (1 − t ) , because u + v = s and u / (u+v) = t , we have that the limits of integrations for s are 0 to ∞ and the limits of integration for t are 0 to 1. Thus produces
Γ
(
z
1
)
Γ
(
z
2
)
=
∫
s
=
0
∞
∫
t
=
0
1
e
−
s
(
s
t
)
z
1
−
1
(
s
(
1
−
t
)
)
z
2
−
1
s
d
t
d
s
=
∫
s
=
0
∞
e
−
s
s
z
1
+
z
2
−
1
d
s
⋅
∫
t
=
0
1
t
z
1
−
1
(
1
−
t
)
z
2
−
1
d
t
=
Γ
(
z
1
+
z
2
)
⋅
B
(
z
1
,
z
2
)
.
{\displaystyle {\begin{aligned}\Gamma (z_{1})\Gamma (z_{2})&=\int _{s=0}^{\infty }\int _{t=0}^{1}e^{-s}(st)^{z_{1}-1}(s(1-t))^{z_{2}-1}s\,dt\,ds\\[6pt]&=\int _{s=0}^{\infty }e^{-s}s^{z_{1}+z_{2}-1}\,ds\cdot \int _{t=0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\,dt\\&=\Gamma (z_{1}+z_{2})\cdot \mathrm {B} (z_{1},z_{2}).\end{aligned}}}
Dividing both sides by
Γ
(
z
1
+
z
2
)
{\displaystyle \Gamma (z_{1}+z_{2})}
gives the desired result.
The stated identity may be seen as a particular case of the identity for the
integral of a convolution . Taking
f
(
u
)
:=
e
−
u
u
z
1
−
1
1
R
+
g
(
u
)
:=
e
−
u
u
z
2
−
1
1
R
+
,
{\displaystyle {\begin{aligned}f(u)&:=e^{-u}u^{z_{1}-1}1_{\mathbb {R} _{+}}\\g(u)&:=e^{-u}u^{z_{2}-1}1_{\mathbb {R} _{+}},\end{aligned}}}
one has:
Γ
(
z
1
)
Γ
(
z
2
)
=
∫
R
f
(
u
)
d
u
⋅
∫
R
g
(
u
)
d
u
=
∫
R
(
f
∗
g
)
(
u
)
d
u
=
B
(
z
1
,
z
2
)
Γ
(
z
1
+
z
2
)
.
{\displaystyle \Gamma (z_{1})\Gamma (z_{2})=\int _{\mathbb {R} }f(u)\,du\cdot \int _{\mathbb {R} }g(u)\,du=\int _{\mathbb {R} }(f*g)(u)\,du=\mathrm {B} (z_{1},z_{2})\,\Gamma (z_{1}+z_{2}).}
Differentiation of the Beta function
We have
∂
∂
z
1
B
(
z
1
,
z
2
)
=
B
(
z
1
,
z
2
)
(
Γ
′
(
z
1
)
Γ
(
z
1
)
−
Γ
′
(
z
1
+
z
2
)
Γ
(
z
1
+
z
2
)
)
=
B
(
z
1
,
z
2
)
(
ψ
(
z
1
)
−
ψ
(
z
1
+
z
2
)
)
,
{\displaystyle {\frac {\partial }{\partial z_{1}}}\mathrm {B} (z_{1},z_{2})=\mathrm {B} (z_{1},z_{2})\left({\frac {\Gamma '(z_{1})}{\Gamma (z_{1})}}-{\frac {\Gamma '(z_{1}+z_{2})}{\Gamma (z_{1}+z_{2})}}\right)=\mathrm {B} (z_{1},z_{2}){\big (}\psi (z_{1})-\psi (z_{1}+z_{2}){\big )},}
∂
∂
z
m
B
(
z
1
,
z
2
,
…
,
z
n
)
=
B
(
z
1
,
z
2
,
…
,
z
n
)
(
ψ
(
z
m
)
−
ψ
(
∑
k
=
1
n
z
k
)
)
,
1
≤
m
≤
n
,
{\displaystyle {\frac {\partial }{\partial z_{m}}}\mathrm {B} (z_{1},z_{2},\dots ,z_{n})=\mathrm {B} (z_{1},z_{2},\dots ,z_{n})\left(\psi (z_{m})-\psi \left(\sum _{k=1}^{n}z_{k}\right)\right),\quad 1\leq m\leq n,}
where
ψ
(
z
)
{\displaystyle \psi (z)}
denotes the
digamma function .
Approximation
Stirling's approximation gives the asymptotic formula
B
(
x
,
y
)
∼
2
π
x
x
−
1
/
2
y
y
−
1
/
2
(
x
+
y
)
x
+
y
−
1
/
2
{\displaystyle \mathrm {B} (x,y)\sim {\sqrt {2\pi }}{\frac {x^{x-1/2}y^{y-1/2}}{({x+y})^{x+y-1/2}}}}
for large x and large y .
If on the other hand x is large and y is fixed, then
B
(
x
,
y
)
∼
Γ
(
y
)
x
−
y
.
{\displaystyle \mathrm {B} (x,y)\sim \Gamma (y)\,x^{-y}.}
Other identities and formulas
The integral defining the beta function may be rewritten in a variety of ways, including the following:
B
(
z
1
,
z
2
)
=
2
∫
0
π
/
2
(
sin
θ
)
2
z
1
−
1
(
cos
θ
)
2
z
2
−
1
d
θ
,
=
∫
0
∞
t
z
1
−
1
(
1
+
t
)
z
1
+
z
2
d
t
,
=
n
∫
0
1
t
n
z
1
−
1
(
1
−
t
n
)
z
2
−
1
d
t
,
=
(
1
−
a
)
z
2
∫
0
1
(
1
−
t
)
z
1
−
1
t
z
2
−
1
(
1
−
a
t
)
z
1
+
z
2
d
t
for any
a
∈
R
≤
1
,
{\displaystyle {\begin{aligned}\mathrm {B} (z_{1},z_{2})&=2\int _{0}^{\pi /2}(\sin \theta )^{2z_{1}-1}(\cos \theta )^{2z_{2}-1}\,d\theta ,\\[6pt]&=\int _{0}^{\infty }{\frac {t^{z_{1}-1}}{(1+t)^{z_{1}+z_{2}}}}\,dt,\\[6pt]&=n\int _{0}^{1}t^{nz_{1}-1}(1-t^{n})^{z_{2}-1}\,dt,\\&=(1-a)^{z_{2}}\int _{0}^{1}{\frac {(1-t)^{z_{1}-1}t^{z_{2}-1}}{(1-at)^{z_{1}+z_{2}}}}dt\qquad {\text{for any }}a\in \mathbb {R} _{\leq 1},\end{aligned}}}
where in the second-to-last identity n is any positive real number. One may move from the first integral to the second one by substituting
t
=
tan
2
(
θ
)
{\displaystyle t=\tan ^{2}(\theta )}
.
The beta function can be written as an infinite sum
[3]
B
(
x
,
y
)
=
∑
n
=
0
∞
(
1
−
x
)
n
(
y
+
n
)
n
!
{\displaystyle \mathrm {B} (x,y)=\sum _{n=0}^{\infty }{\frac {(1-x)_{n}}{(y+n)\,n!}}}
(where
(
x
)
n
{\displaystyle (x)_{n}}
is the
rising factorial )
and as an infinite product
B
(
x
,
y
)
=
x
+
y
x
y
∏
n
=
1
∞
(
1
+
x
y
n
(
x
+
y
+
n
)
)
−
1
.
{\displaystyle \mathrm {B} (x,y)={\frac {x+y}{xy}}\prod _{n=1}^{\infty }\left(1+{\dfrac {xy}{n(x+y+n)}}\right)^{-1}.}
The beta function satisfies several identities analogous to corresponding identities for binomial coefficients, including a version of
Pascal's identity
B
(
x
,
y
)
=
B
(
x
,
y
+
1
)
+
B
(
x
+
1
,
y
)
{\displaystyle \mathrm {B} (x,y)=\mathrm {B} (x,y+1)+\mathrm {B} (x+1,y)}
and a simple recurrence on one coordinate:
B
(
x
+
1
,
y
)
=
B
(
x
,
y
)
⋅
x
x
+
y
,
B
(
x
,
y
+
1
)
=
B
(
x
,
y
)
⋅
y
x
+
y
.
{\displaystyle \mathrm {B} (x+1,y)=\mathrm {B} (x,y)\cdot {\dfrac {x}{x+y}},\quad \mathrm {B} (x,y+1)=\mathrm {B} (x,y)\cdot {\dfrac {y}{x+y}}.}
[4]
The positive integer values of the beta function are also the partial derivatives of a 2D function: for all nonnegative integers
m
{\displaystyle m}
and
n
{\displaystyle n}
,
B
(
m
+
1
,
n
+
1
)
=
∂
m
+
n
h
∂
a
m
∂
b
n
(
0
,
0
)
,
{\displaystyle \mathrm {B} (m+1,n+1)={\frac {\partial ^{m+n}h}{\partial a^{m}\,\partial b^{n}}}(0,0),}
where
h
(
a
,
b
)
=
e
a
−
e
b
a
−
b
.
{\displaystyle h(a,b)={\frac {e^{a}-e^{b}}{a-b}}.}
The Pascal-like identity above implies that this function is a solution to the
first-order partial differential equation
h
=
h
a
+
h
b
.
{\displaystyle h=h_{a}+h_{b}.}
For
x
,
y
≥
1
{\displaystyle x,y\geq 1}
, the beta function may be written in terms of a
convolution involving the
truncated power function
t
↦
t
+
x
{\displaystyle t\mapsto t_{+}^{x}}
:
B
(
x
,
y
)
⋅
(
t
↦
t
+
x
+
y
−
1
)
=
(
t
↦
t
+
x
−
1
)
∗
(
t
↦
t
+
y
−
1
)
{\displaystyle \mathrm {B} (x,y)\cdot \left(t\mapsto t_{+}^{x+y-1}\right)={\Big (}t\mapsto t_{+}^{x-1}{\Big )}*{\Big (}t\mapsto t_{+}^{y-1}{\Big )}}
Evaluations at particular points may simplify significantly; for example,
B
(
1
,
x
)
=
1
x
{\displaystyle \mathrm {B} (1,x)={\dfrac {1}{x}}}
and
B
(
x
,
1
−
x
)
=
π
sin
(
π
x
)
,
x
∉
Z
{\displaystyle \mathrm {B} (x,1-x)={\dfrac {\pi }{\sin(\pi x)}},\qquad x\not \in \mathbb {Z} }
[5]
By taking
x
=
1
2
{\displaystyle x={\frac {1}{2}}}
in this last formula, it follows that
Γ
(
1
/
2
)
=
π
{\displaystyle \Gamma (1/2)={\sqrt {\pi }}}
.
Generalizing this into a bivariate identity for a product of beta functions leads to:
B
(
x
,
y
)
⋅
B
(
x
+
y
,
1
−
y
)
=
π
x
sin
(
π
y
)
.
{\displaystyle \mathrm {B} (x,y)\cdot \mathrm {B} (x+y,1-y)={\frac {\pi }{x\sin(\pi y)}}.}
Euler's integral for the beta function may be converted into an integral over the
Pochhammer contour C as
(
1
−
e
2
π
i
α
)
(
1
−
e
2
π
i
β
)
B
(
α
,
β
)
=
∫
C
t
α
−
1
(
1
−
t
)
β
−
1
d
t
.
{\displaystyle \left(1-e^{2\pi i\alpha }\right)\left(1-e^{2\pi i\beta }\right)\mathrm {B} (\alpha ,\beta )=\int _{C}t^{\alpha -1}(1-t)^{\beta -1}\,dt.}
This Pochhammer contour integral converges for all values of α and β and so gives the
analytic continuation of the beta function.
Just as the gamma function for integers describes
factorials , the beta function can define a
binomial coefficient after adjusting indices:
(
n
k
)
=
1
(
n
+
1
)
B
(
n
−
k
+
1
,
k
+
1
)
.
{\displaystyle {\binom {n}{k}}={\frac {1}{(n+1)\,\mathrm {B} (n-k+1,k+1)}}.}
Moreover, for integer n , Β can be factored to give a closed form interpolation function for continuous values of k :
(
n
k
)
=
(
−
1
)
n
n
!
⋅
sin
(
π
k
)
π
∏
i
=
0
n
(
k
−
i
)
.
{\displaystyle {\binom {n}{k}}=(-1)^{n}\,n!\cdot {\frac {\sin(\pi k)}{\pi \displaystyle \prod _{i=0}^{n}(k-i)}}.}
Reciprocal beta function
The reciprocal beta function is the
function about the form
f
(
x
,
y
)
=
1
B
(
x
,
y
)
{\displaystyle f(x,y)={\frac {1}{\mathrm {B} (x,y)}}}
Interestingly, their integral representations closely relate as the
definite integral of
trigonometric functions with product of its power and
multiple-angle :
[6]
∫
0
π
sin
x
−
1
θ
sin
y
θ
d
θ
=
π
sin
y
π
2
2
x
−
1
x
B
(
x
+
y
+
1
2
,
x
−
y
+
1
2
)
{\displaystyle \int _{0}^{\pi }\sin ^{x-1}\theta \sin y\theta ~d\theta ={\frac {\pi \sin {\frac {y\pi }{2}}}{2^{x-1}x\mathrm {B} \left({\frac {x+y+1}{2}},{\frac {x-y+1}{2}}\right)}}}
∫
0
π
sin
x
−
1
θ
cos
y
θ
d
θ
=
π
cos
y
π
2
2
x
−
1
x
B
(
x
+
y
+
1
2
,
x
−
y
+
1
2
)
{\displaystyle \int _{0}^{\pi }\sin ^{x-1}\theta \cos y\theta ~d\theta ={\frac {\pi \cos {\frac {y\pi }{2}}}{2^{x-1}x\mathrm {B} \left({\frac {x+y+1}{2}},{\frac {x-y+1}{2}}\right)}}}
∫
0
π
cos
x
−
1
θ
sin
y
θ
d
θ
=
π
cos
y
π
2
2
x
−
1
x
B
(
x
+
y
+
1
2
,
x
−
y
+
1
2
)
{\displaystyle \int _{0}^{\pi }\cos ^{x-1}\theta \sin y\theta ~d\theta ={\frac {\pi \cos {\frac {y\pi }{2}}}{2^{x-1}x\mathrm {B} \left({\frac {x+y+1}{2}},{\frac {x-y+1}{2}}\right)}}}
∫
0
π
2
cos
x
−
1
θ
cos
y
θ
d
θ
=
π
2
x
x
B
(
x
+
y
+
1
2
,
x
−
y
+
1
2
)
{\displaystyle \int _{0}^{\frac {\pi }{2}}\cos ^{x-1}\theta \cos y\theta ~d\theta ={\frac {\pi }{2^{x}x\mathrm {B} \left({\frac {x+y+1}{2}},{\frac {x-y+1}{2}}\right)}}}
Incomplete beta function
The incomplete beta function , a generalization of the beta function, is defined as
[7]
[8]
B
(
x
;
a
,
b
)
=
∫
0
x
t
a
−
1
(
1
−
t
)
b
−
1
d
t
.
{\displaystyle \mathrm {B} (x;\,a,b)=\int _{0}^{x}t^{a-1}\,(1-t)^{b-1}\,dt.}
For x = 1 , the incomplete beta function coincides with the complete beta function. The relationship between the two functions is like that between the gamma function and its generalization the
incomplete gamma function . For positive integer a and b , the incomplete beta function will be a polynomial of degree a + b - 1 with rational coefficients.
The regularized incomplete beta function (or regularized beta function for short) is defined in terms of the incomplete beta function and the complete beta function:
I
x
(
a
,
b
)
=
B
(
x
;
a
,
b
)
B
(
a
,
b
)
.
{\displaystyle I_{x}(a,b)={\frac {\mathrm {B} (x;\,a,b)}{\mathrm {B} (a,b)}}.}
The regularized incomplete beta function is the
cumulative distribution function of the
beta distribution , and is related to the
cumulative distribution function
F
(
k
;
n
,
p
)
{\displaystyle F(k;\,n,p)}
of a
random variable X following a
binomial distribution with probability of single success p and number of Bernoulli trials n :
F
(
k
;
n
,
p
)
=
Pr
(
X
≤
k
)
=
I
1
−
p
(
n
−
k
,
k
+
1
)
=
1
−
I
p
(
k
+
1
,
n
−
k
)
.
{\displaystyle F(k;\,n,p)=\Pr \left(X\leq k\right)=I_{1-p}(n-k,k+1)=1-I_{p}(k+1,n-k).}
Properties
I
0
(
a
,
b
)
=
0
I
1
(
a
,
b
)
=
1
I
x
(
a
,
1
)
=
x
a
I
x
(
1
,
b
)
=
1
−
(
1
−
x
)
b
I
x
(
a
,
b
)
=
1
−
I
1
−
x
(
b
,
a
)
I
x
(
a
+
1
,
b
)
=
I
x
(
a
,
b
)
−
x
a
(
1
−
x
)
b
a
B
(
a
,
b
)
I
x
(
a
,
b
+
1
)
=
I
x
(
a
,
b
)
+
x
a
(
1
−
x
)
b
b
B
(
a
,
b
)
∫
B
(
x
;
a
,
b
)
d
x
=
x
B
(
x
;
a
,
b
)
−
B
(
x
;
a
+
1
,
b
)
B
(
x
;
a
,
b
)
=
(
−
1
)
a
B
(
x
x
−
1
;
a
,
1
−
a
−
b
)
{\displaystyle {\begin{aligned}I_{0}(a,b)&=0\\I_{1}(a,b)&=1\\I_{x}(a,1)&=x^{a}\\I_{x}(1,b)&=1-(1-x)^{b}\\I_{x}(a,b)&=1-I_{1-x}(b,a)\\I_{x}(a+1,b)&=I_{x}(a,b)-{\frac {x^{a}(1-x)^{b}}{a\mathrm {B} (a,b)}}\\I_{x}(a,b+1)&=I_{x}(a,b)+{\frac {x^{a}(1-x)^{b}}{b\mathrm {B} (a,b)}}\\\int B(x;a,b)\mathrm {d} x&=xB(x;a,b)-B(x;a+1,b)\\\mathrm {B} (x;a,b)&=(-1)^{a}\mathrm {B} \left({\frac {x}{x-1}};a,1-a-b\right)\end{aligned}}}
Continued fraction expansion
The
continued fraction expansion
B
(
x
;
a
,
b
)
=
x
a
(
1
−
x
)
b
a
(
1
+
d
1
1
+
d
2
1
+
d
3
1
+
d
4
1
+
⋯
)
{\displaystyle \mathrm {B} (x;\,a,b)={\frac {x^{a}(1-x)^{b}}{a\left(1+{\frac {{d}_{1}}{1+}}{\frac {{d}_{2}}{1+}}{\frac {{d}_{3}}{1+}}{\frac {{d}_{4}}{1+}}\cdots \right)}}}
with odd and even coefficients respectively
d
2
m
+
1
=
−
(
a
+
m
)
(
a
+
b
+
m
)
x
(
a
+
2
m
)
(
a
+
2
m
+
1
)
{\displaystyle {d}_{2m+1}=-{\frac {(a+m)(a+b+m)x}{(a+2m)(a+2m+1)}}}
d
2
m
=
m
(
b
−
m
)
x
(
a
+
2
m
−
1
)
(
a
+
2
m
)
{\displaystyle {d}_{2m}={\frac {m(b-m)x}{(a+2m-1)(a+2m)}}}
converges rapidly when
x
{\displaystyle x}
is not close to 1. The
4
m
{\displaystyle 4m}
and
4
m
+
1
{\displaystyle 4m+1}
convergents are less than
B
(
x
;
a
,
b
)
{\displaystyle \mathrm {B} (x;\,a,b)}
, while the
4
m
+
2
{\displaystyle 4m+2}
and
4
m
+
3
{\displaystyle 4m+3}
convergents are greater than
B
(
x
;
a
,
b
)
{\displaystyle \mathrm {B} (x;\,a,b)}
.
For
x
>
a
+
1
a
+
b
+
2
{\displaystyle x>{\frac {a+1}{a+b+2}}}
, the function may be evaluated more efficiently using
B
(
x
;
a
,
b
)
=
B
(
a
,
b
)
−
B
(
1
−
x
;
b
,
a
)
{\displaystyle \mathrm {B} (x;\,a,b)=\mathrm {B} (a,b)-\mathrm {B} (1-x;\,b,a)}
.
[8]
Multivariate beta function
The beta function can be extended to a function with more than two arguments:
B
(
α
1
,
α
2
,
…
α
n
)
=
Γ
(
α
1
)
Γ
(
α
2
)
⋯
Γ
(
α
n
)
Γ
(
α
1
+
α
2
+
⋯
+
α
n
)
.
{\displaystyle \mathrm {B} (\alpha _{1},\alpha _{2},\ldots \alpha _{n})={\frac {\Gamma (\alpha _{1})\,\Gamma (\alpha _{2})\cdots \Gamma (\alpha _{n})}{\Gamma (\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n})}}.}
This multivariate beta function is used in the definition of the
Dirichlet distribution . Its relationship to the beta function is analogous to the relationship between
multinomial coefficients and binomial coefficients. For example, it satisfies a similar version of Pascal's identity:
B
(
α
1
,
α
2
,
…
α
n
)
=
B
(
α
1
+
1
,
α
2
,
…
α
n
)
+
B
(
α
1
,
α
2
+
1
,
…
α
n
)
+
⋯
+
B
(
α
1
,
α
2
,
…
α
n
+
1
)
.
{\displaystyle \mathrm {B} (\alpha _{1},\alpha _{2},\ldots \alpha _{n})=\mathrm {B} (\alpha _{1}+1,\alpha _{2},\ldots \alpha _{n})+\mathrm {B} (\alpha _{1},\alpha _{2}+1,\ldots \alpha _{n})+\cdots +\mathrm {B} (\alpha _{1},\alpha _{2},\ldots \alpha _{n}+1).}
Applications
The beta function is useful in computing and representing the
scattering amplitude for
Regge trajectories . Furthermore, it was the first known
scattering amplitude in
string theory , first conjectured by
Gabriele Veneziano . It also occurs in the theory of the
preferential attachment process, a type of stochastic
urn process . The beta function is also important in statistics, e.g. for the
Beta distribution and
Beta prime distribution . As briefly alluded to previously, the beta function is closely tied with the
gamma function and plays an important role in
calculus .
Software implementation
Even if unavailable directly, the complete and incomplete beta function values can be calculated using functions commonly included in
spreadsheet or
computer algebra systems .
In
Microsoft Excel , for example, the complete beta function can be computed with the
GammaLn
function (or special.gammaln
in
Python's
SciPy package):
Value = Exp(GammaLn(a) + GammaLn(b) − GammaLn(a + b))
This result follows from the properties
listed above .
The incomplete beta function cannot be directly computed using such relations and other methods must be used. In
GNU Octave , it is computed using a
continued fraction expansion.
The incomplete beta function has existing implementation in common languages. For instance, betainc
(incomplete beta function) in
MATLAB and
GNU Octave , pbeta
(probability of beta distribution) in
R , or special.betainc
in
SciPy compute the
regularized incomplete beta function —which is, in fact, the cumulative beta distribution—and so, to get the actual incomplete beta function, one must multiply the result of betainc
by the result returned by the corresponding beta
function. In
Mathematica , Beta[x, a, b]
and BetaRegularized[x, a, b]
give
B
(
x
;
a
,
b
)
{\displaystyle \mathrm {B} (x;\,a,b)}
and
I
x
(
a
,
b
)
{\displaystyle I_{x}(a,b)}
, respectively.
See also
References
^
a
b
c Davis, Philip J. (1972), "6. Gamma function and related functions", in
Abramowitz, Milton ;
Stegun, Irene A. (eds.),
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , New York:
Dover Publications , p. 258,
ISBN
978-0-486-61272-0 . Specifically, see 6.2 Beta Function.
^ Artin, Emil,
The Gamma Function (PDF) , pp. 18–19, archived from
the original (PDF) on 2016-11-12, retrieved 2016-11-11
^
Beta function : Series representations (Formula 06.18.06.0007)
^ Mäklin, Tommi (2022),
Probabilistic Methods for High-Resolution Metagenomics (PDF) , Series of publications A / Department of Computer Science, University of Helsinki, Helsinki: Unigrafia, p. 27,
ISBN
978-951-51-8695-9 ,
ISSN
2814-4031
^
"Euler's Reflection Formula - ProofWiki" , proofwiki.org , retrieved 2020-09-02
^ Paris, R. B. (2010),
"Beta Function" , in
Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),
NIST Handbook of Mathematical Functions , Cambridge University Press,
ISBN
978-0-521-19225-5 ,
MR
2723248 .
^ Zelen, M.; Severo, N. C. (1972), "26. Probability functions", in
Abramowitz, Milton ;
Stegun, Irene A. (eds.),
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , New York:
Dover Publications , pp.
944 ,
ISBN
978-0-486-61272-0
^
a
b Paris, R. B. (2010),
"Incomplete beta functions" , in
Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),
NIST Handbook of Mathematical Functions , Cambridge University Press,
ISBN
978-0-521-19225-5 ,
MR
2723248 .
Askey, R. A. ; Roy, R. (2010),
"Beta function" , in
Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),
NIST Handbook of Mathematical Functions , Cambridge University Press,
ISBN
978-0-521-19225-5 ,
MR
2723248 .
Press, W. H.; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007),
"Section 6.1 Gamma Function, Beta Function, Factorials" , Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press,
ISBN
978-0-521-88068-8 , archived from
the original on 2021-10-27, retrieved 2011-08-09
External links