  A low-frequency gravitational-wave background (GWB) from the cosmic merger
history of supermassive black holes is expected to be detected in the next few
years by pulsar timing arrays. A GWB induces distinctive correlations in the
pulsar residuals --- the expected arrival time of the pulse less its actual
arrival time. Simplifying assumptions are made in order to write an analytic
expression for this correlation function, called the Hellings and Downs curve
for an isotropic GWB, which depends on the angular separation of the pulsar
pairs, the gravitational-wave frequency considered, and the distance to the
pulsars. This is called the short-wavelength approximation, which we prove here
rigorously and analytically for the first time.
