  We present a method for constructing global analytical expressions that
approximate a function over its entire range. These approximations not only
mirror the original function as accurately as desired, but are purposefully
created to possess features that the original function lacks. This is
particularly useful for functions that lack closed form and are defined by
integrals or infinite series. Replacing these definitions with simple
analytical expressions enables in-depth qualitative analysis and replaces the
current methods of evaluation. We demonstrate this procedure by providing
replacements for a variety of pivotal functions in physics and cosmology
including the pressure and density of quantum gas, the one-loop correction in
thermal field theory, common polylog functions, and the error function.
