  We theoretically study the magnetic excitations in a frustrated two-leg
spin-ladder system, in which antiferromagnetic exchange interactions act on the
nearest-neighbor and next-nearestneighbor bonds in the leg direction, and on
the nearest-neighbor bonds in the rung direction. A dynamical spin correlation
function at zero temperature is calculated by using the dynamical
density-matrix renormalization-group method for possible magnetic phases, i.e.,
columnar-dimer and rung-singlet phases. The columnar-dimer phase is
characterized by multi-spinon excitations with spin gap, while the rung-singlet
phase is dominated by the triplet excitation in the rung direction. It is found
that a major difference between these two phases appears in the spectral weight
of magnetic excitations, in particular, of the bonding and anti-bonding modes
in the rung direction. Therefore, we can distinguish one phase from the other
by the difference of the spectral weight. Furthermore, we examine frustration
effect on both modes in the rung-singlet phase with a perturbation theory from
the strong coupling limit. It is shown that the anti-bonding mode is stable
against the frustration, and a wavenumber with minimum excitation energy is
shifted from the commensurate to incommensurate ones. On the other hand, the
bonding mode is merged into the continuum excitation of multiple triplet
excitations by increasing the frustration. By comparing our results with
inelastic neutron scattering experiments for BiCu$_2$PO$_6$, the magnitude of
the magnetic exchange interactions and the ground state will be determined.
