Solutions to Fractional Gamma Function Problems

1. G(x + 1) = x G (x)

Then: 


G (3/2) = G ( ½ +  1) = ½ G (½) =  Öp/2


2.   G (-0.70) = ?

Use:  G(x) = G(x + 1) = x G (x)


so that: G (-0.70) = G (-0.70 + 1) =  -0.70 G (-0.70)

Rewrite this as:  

G (-0.70)  =  G (-0.70 + 1) /  - 0.70  = G (0.30)/ - 0.70

But:    

G (0.30) = 2.992   

so  G (0.30) / - 0.70 = 2.992/ -0.70 = -4.274

  3.  We use the form:
 
G (n + ½) = (2n - 1)! (p)^1/2 / 2ⁿ n!

To find:   G (3/2)

Here: n = 1,  i.e. form is (½ + 1)= 3/2,  so that:

(2n - 1)! (p)^1/2 / 2ⁿ n!   =  (1) (p)^1/2  / 2¹  1!    =  (p)^1/2  / 2  =    Öp/2

Which is the same answer as (1) obtained in a different way.


4.   G (5/2)  = G ( 3/2 +  1) = 3/2  G (3/2)  

(Since G(x + 1) = x G (x)  )

And we know:

G (3/2) =  Öp/2

Therefore:

G (5/2)  =  3/2   (Öp/2)  =   3Öp/ 4