  We discuss the creation of electron-positron cascades in the context of
pulsar polar cap acceleration models and derive several useful analytic and
semi-analytic results for the spatial extent and energy response of the
cascade. Instead of Monte Carlo simulations, we use an integro- differential
equation which describes the development of the cascade energy spectrum in one
space dimension quite well, when it is compared to existing Monte Carlo models.
We reduce this full equation to a single integral equation, from which we can
derive useful results, such as the energy loss between successive generations
of photons and the spectral index of the response. We find that a simple
analytic formula represents the pair cascade multiplicity quite well, provided
that the magnetic field is below 10^12 Gauss, and that an only slightly more
complex formula matches the numerically-calculated cascade at all other field
strengths. Using these results, we find that cascades triggered by gamma rays
emitted through inverse Compton scattering of thermal photons from the neutron
star's surface, both resonant and non-resonant, are important for the dynamics
of the polar cap region in many pulsars.
