Main Index Mathematical Analysis Special Functions
  Subject Index
comment on the page

Beta function

Euler Beta function or Euler function of the first kind (this name was introduced by Legendre) is defined by the integral

B(a, b) = Underoverscript[∫, 0, arg3] x^(a - 1)(1 - x)^(b - 1) d x (1)

which converges if and only if typeset structure, typeset structure.

To compute values of  typeset structure go to .

The graph of Beta function

[Graphics:HTMLFiles/BetaFunction_5.gif]

Substitution typeset structure immediately gives that

B(a, b) = B(b, a) .(2)

Integration by parts in Equation 1 yields

B(a, b) = (b - 1)/a B(a, b - 1) - (b - 1)/a B(a, b)

and consequently

B(a, b) = (b - 1)/(a + b - 1) B(a, b - 1) = (a - 1)/(a + b - 1) B(a - 1, b) ,(3)

where the last equation is a consequence the symmetry relation Equation 2.

If typeset structure is a positive integer then Equation 3 implies

B(a, n) = (n - 1)/(a + n - 1) · (n - 2)/(a + n - 2) ... 1/(a + 1) · B(a, 1) .

Since typeset structure,

B(a, n) = (n - 1) !/(a(a + 1) (a + 2) ...(a + n - 1)) = (n - 1) !/(a) _ n,(4)

where typeset structure is the Pochhammer symbol . Thus if both typeset structure and typeset structure are positive integers, then

B(n, m) = ((n - 1) ! (m - 1) !)/(n + m - 1) ! .(5)

Another expression for Beta function we get using the substitution typeset structure in Equation 1

B(a, b) = Underoverscript[∫, 0, arg3] y^(a - 1)/(1 + y)^(a + b) d y = 2 Underoverscript[∫, 0, arg3] y^(2 a - 1)/(1 + y^2)^(a + b) d y .(6)

If typeset structure then the substitution typeset structure in first integral of  Equation 6 gives

B(a, 1 - a) = Underoverscript[∫, 0, arg3] y^(a - 1)/(1 + y) d y = π/(sin πa)   ... sp;              0 < a < 1.(7)

In particular

B(1/2, 1/2) = π .(8)

From Equation 6 there follows

B(a, b) Γ(a + b) = Underoverscript[∫, 0, arg3] (Γ(a + b) y^(a - 1))/(1 + y)^(a ... 7;, 0, arg3] e^(-x) x^(b - 1) d x Underoverscript[∫, 0, arg3] e^(-x y)(x y)^(a - 1) d (xy),

what gives an extension of Equation 5 to non-integral values of arguments

B(a, b) = (Γ(a) Γ(b))/Γ(a + b)(9)

This can be also proved using Laplace transform [1] . We have

L {t^(k - 1)} = Γ(k) s^(-k)            for         k > 0 .(10)

Using the convolution integral and the substitution typeset structure we obtain

Γ(p) s^(-p) Γ(q) s^(-q) = L {Underoverscript[∫, 0, arg3] τ^(p - 1)(t - &ta ... - 1) Underoverscript[∫, 0, arg3] x^(p - 1)(1 - x)^(q - 1) d x} = L {t^(p + q - 1) B(p, q)} .

Equation 10  shows that  typeset structure for typeset structure and typeset structure, and Equation 9 is again proved.

We also have

B(a, b) = 1/a Underoverscript[∑, k = 0, arg3] (-1)^k (b(b - 1) ...(b - k))/(k ! (a + k)),        b > 0,(11)
B(a, b) = Underoverscript[∏, k = 0, arg3] k(a + b + k)/((a + k) (b + k)),(12)
B(a, b) = 2 Underoverscript[∫, 0, arg3] sin^(2 a - 1) φ cos^(2 b - 1) φ d φ,(13)
B(a, b) = 2 Underoverscript[∫, 0, arg3] sin^(2 a ) φ cos^(2 b) φ d φ , &n ...              a > -1/2, b > -1/2,(14)
B(a, b) = Underoverscript[∫, 0, arg3] (x^(a - 1) + x^(b - 1))/(1 + x)^(a + b) d x,               a > 0, b > 0,(15)
B(a, a) = 1/2^(2 n - 1) Underoverscript[∫, 0, arg3] (1 - x)^(a - 1)/x^(1/2) d x .(16)

References

[1]  Brown, J. W. (1961, Feb.). The Beta-Gamma Function Identity. Amer. Math. Monthly, 68, 165.

Cite this web-page as:

Štefan Porubský: Beta Function.

Page created  .