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)

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)