Hurwitz Zeta function

HZ.ID.1:  Hurwitz Zeta- Gamma relation

Proof

 

 (1)

From (), the multiplication of (1) with gamma integral definition

 (2)

change of variable

 (3)

substituting (3) into (2)

 (4)

 (5)

 (6)

using geometric series

 (7)

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)