Saturday, September 8, 2012

Dirichlet Beta - Hurwitz zeta relation

DBHZ.ID.1: Dirichlet Beta - Hurwitz zeta relation

 
Proof
From  DBG.ID.1 , the 







 Multiply both numerator and denominator of integral by




 (1)
The integral representation of Hurwitz zeta function:



(2)





change of variable x=2t , dx=2dt





(3)

similarly


(4)



(5)




(6)

substituting (6) into (5)



(7)


(8)


change of variable t= 2x , dt=2dx




(9)

from (1) and (9) , Dirichlet Beta - Hurwitz zeta relation is given as by: 


(10)







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