In this chapter we will consider methods of constructing asymptotic approximations of integrals. Specifically we will study integrals of the form

To motivate this study consider the one-dimensional diffusion convection equation

d2tP dtP - v dXz + U(X} dX = F(X},

where tP may represent various physical properties of the medium such as temperature, chemical concentration or fluid momentum. The property tP is transported by the medium with speed U(X) and diffuses through the medium whose diffusivity is given by v. The right-hand side, F(x), represents the source of the quantity tP·

Multiplying throughout by the integrating factor exp(-UX/v) allows the equation to be written in the following form,

-v~(e-uxt• dtP) = eVXt• F dX 'dX .. Thus on integrating we have

dtP = - ~eUX!• re-UXI•FdX dX v J' .