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)
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