Looking Again At The Beta - And Gamma - Functions

 We first consider the integral:

ò ¹ _o   x  ^u-1  (1 – x)  ^v-1  dx =  b (u,v) 

For Re u  >  0,   Re v >  0

Which is known as the Beta function.

This function is symmetric so can also be written:

ò ¹ _o   y  ^v-1  (1 – y)  ^u-1  dx =  b (v, u)

This  can be expressed in trigonometric form by writing:

x  =    sin ² q 

Thence:

ò ^p/2 _o    sin  ^2u-2 q  cos  ^2v-2 q  2 sin  q cos q  dq   =  b (u,v) 

Or:

b (u,v)   = 2  ò ^p/2 _o    sin  ^2u-1 q  cos  ^2v-1 q   dq

The earlier expressions for  b (u,v)  and b (v, u)  represent analytic functions in each of the complex variables u and v.  One can then introduce a new variable of integration:  w =  x / (1 -  x)  so the original integral becomes:

b (u,v)   =    ò ^¥ _o    w ^u-1  dw / (1 + w) ^u+v  

For Re u  >  0,   Re v >  0   

Using suitable substitutions, we find that:

  (1 + w) ^u+v    = 

 1/ G(u + v)  ò ^¥ _o   exp (-1 + w) x^u+v-1  dx

Where   G(u + v)  is the gamma function

Substituting the preceding expression (for  (1 + w) ^u+v )

into:   b (u,v)   =    ò ^¥ _o    w ^u-1  dw / (1 + w) ^u+v 

We get:

 b (u,v)   =   G(u) / G(u + v)  ò ^¥ _o   e ^{- x}   x^v-1  dx =  G(u) G(v)  / G(u + v)

A useful identity.

Now, to fix ideas, consider the Beta function b(3, 4):

b(3, 4) = G (3) G (4)/ G (3 + 4) = (2) (6)/ G (7) = 12 / G (7)

Recall:   G (a) = (a - 1)!

So:  G (3) = (3 - 1)!  =  2!  =  2 x 1 = 2

G (4) = (4 - 1)!  =  3!  =  3 x 2 x 1 = 6

Then  b(3, 4) =  12/ (7 - 1)! = 12/ (6)!

=   12/  (6 x 5 x 4 x 3 x 2 x 1)  =   12/ 720 = 1/ 60

Problems for Math Wizards:

1) The error function erf x

 =  (2 /Ö  p )  ò ^¥ _o   exp  ( -x ²) dx

The gamma function:  

G (½ ) =  ò ^¥ _o   x ^-1/2  e ^{- x}    dx

Write the relationship between erf x and G (½)

2)  Given:  ò ¹ _o   x ⁿ  dx /Ö ( 1 -  x ²)   

Find an integral expression for the Beta function:     ½ b(n +  ½,   ½)

Hint:  Let x  =    sin  q 

3)  Astrophysicist Brian Greene in an episode of 'The Elegant Universe', wrote out a form of the Beta function used in string theory as:  

b(p, q) = G (p) G (q)/ G (p + q)

Where:   p  = [1- a(s)]    and q  =   [1 - a(t)]

Are string theory parameters

If:   a(s)  =   ln e/ 20   and   a(t)  =    p ln e/ 2

Find the applicable Beta function.