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

Proof

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)