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Applying the equations of motion together with corresponding boundaryconditions of bulk profiles at infrared and ultraviolet branes, we verify somelemmas on the eigenvalues of Kaluze-Klein modes in framework of warped extradimension with the custodial symmetry $SU(3)_c\times SU(2)_L\timesSU(2)_R\times U(1)_X\times P_{LR}$. Using the lemmas and performing properlyanalytic extensions of bulk profiles, we present the sufficient condition for aconvergent series of Kaluze-Klein excitations and sum over the series throughthe residue theorem. The method can also be applied to sum over the infiniteseries of Kaluze-Klein excitations in unified extra dimension. Additional, weanalyze the possible connection between the propagators in five dimensionalfull theory and the product of bulk profiles with corresponding propagators ofexciting Kaluze-Klein modes in four dimensional effective theory, and recoversome relations presented in literature for warped and unified extra dimensionsrespectively. As an example, we demonstrate that the corrections from neutralHiggs to the Wilson coefficients of relevant operators for $B\rightarrowX_s\gamma$ contain the suppression factor $m_b^3m_s/m_{_{\rm w}}^4$ comparingwith that from other sectors, thus can be neglected safely.

Residue theorem and summing over Kaluza-Klein excitations