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)