Thursday, February 16, 2023

Revisiting The Fractional Gamma Function

 The fractional gamma function  is an offshoot of how the regular gamma function works. One of the more useful formulas for the latter, introduced in several earlier posts e.g.

 is:


G(x + 1) = x G (x)


This will also be found very useful in working with fractional Gamma functions, as I will show in this article. Most solutions of fractional G (x) entail already knowing at least one basic form, usually obtained from a special integral.

For example, working with most fractionals we make use of the basic integral that generates:  G (½) 

G (½)  =   ò¥ o  [t-1/2 exp(-t)] dt

The resulting integral yieldsÖp
 
Now let's see how to obtain G (-½):

From the basic Gamma function formula (letting x = -½) :

G (-½) = G (-½ + 1) = -½ G (-½) 

Or:

G (-½) = -2G (½) = -2 Öp 

Another form of the fractional gamma function appears as decimals - which are merely another form of fraction:

Say you wish to obtain G (-0.30)

In this case, one is assumed to know the basic Gamma function G (1.70) = 0.90864

Then from the gamma function formula given earlier:

G (-0.30) = G (1 - 0.30) = -0.30 G (-0.30)

G (0.70)/ (-0.30) = G (-0.30)

But: G (0.70) = G (1.70)/ 0.70 = (0.90864)/ 0.70 = 1.29805



So: G (-0.30) = G (0.70)/ -0.30 = 1.29805/ -0.30 = -4.32683


Fractional sequences can also come into play, e.g.

Find: G (n + ½):    G (n + ½) = (n- ½) G (n – ½)

 

Since: we use x = (n – ½) in: G (x + 1) = xG (x)

 

G ([n – ½] + 1) = (n – ½) G (n – ½) 

  Þ G (n + ½) = (n – ½) G (n – ½)

We can go further, focusing on treating the right hand side:

(n – ½) G (n – ½) = (n – ½) (n – 3/2) G (n –3/2)

= (n - ½)(n - 3/2) . .. . .3/2·½·G (½)


But, G (½) =  Öp, so:

 

G (n + ½) = (2n -1)/2 · (2n -3)/2 . . .. 3/2· ½· (p)1/2

  Further factoring and additional algebra yields:


G (n + ½) = (2n - 1)! (p)1/2 / 2n n! 

This is left as an exercise for the ambitious reader!


Problems for the Math Maven:

1) Find: G (3/2)

2) Find: G (-0.70)

3) Use the form: G (n + ½) = (2n - 1)! (p)1/2 / 2n n!

and show it is equal to the value for G (3/2)  you obtained in (1)


4) Hence, or otherwise, compute: G (5/2)

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