Friday, May 27, 2011

Beta Function

firstly , I recommend you  to read these  Beta function   to have fundamental understanding of the Gamma function.


G.ID.1:  integral representation

 

G.ID.1: Recurrence

 

 


Proof
(1)
(2)
(3)
 
 (4)
(5)


integrating by parts of (1) 


 
 (6)
from (1) and (6) 
 (7)
from (5) and (7)  


 (8)
from (7) and (8),



 (9)





 (10)
 (11)

G.ID.2:  integral representation


given

proof
(1)
substitute, 
(2)


(3)
(4)


(5)

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