  The Dirac delta function is widely used in many areas of physics and
mathematics. Here we consider the generalization of a Dirac delta function to
allow the use of complex arguments. We show that the properties of a
generalized delta function are very different from those of a Dirac delta
function and that they behave more like a pole in the complex plane. We use the
generalized delta function to derive the Glauber-Sudarshan P-function for a
Schr\"odinger cat state in a surprisingly simple form. Aside from their
potential applications in classical electromagnetism and quantum optics, these
results provide insight into the ability of the diagonal P-function to describe
density operators with off-diagonal elements.
