Saturday, June 2, 2012

Dirichlet Beta - Gamma

DBG.ID.1: Integral representation



given

Dirchelet Beta  :

 
 (1)
 
 (2)

 (3)
replacing (2) with (3)

 (4)

 (5)
using geometric series

 (6)

 (7)


DBG.ID.2: Integral representation




proof

multiply both sides by s
 (8)


 (9)



 (10)


 (11)


DBG.ID.3:Recurrence Relation


 

given

from  DBG.ID.1
 
 (1)


  (2)
 from  DBG.ID.2
 
  (3)

 (3) - (2)  results in : 


 
   (4)
 

   (5)

DBG.ID.5: Integral representation by natural log




from  DBG.ID.1


 (1)

Change of variable 


Therefore,
(2)

(3)
if s=2


(4)



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