Inversion of the fundamental isentropic expansion equations in variable area duct flow

The isentropic flow equations relating the thermodynamic pressures, temperatures, and densities to their stagnation properties are solved in terms of the area ratio and Mach number. These relationships are inverted asymptotically and presented to arbitrary order. Both subsonic and supersonic branches of the solution are systematically identified and produced separately. Two types of recursive formulations are provided, with one exhibiting a universal character by virtue of its applicability to all three properties under consideration. In the case of the subsonic branch, the asymptotic series expansion is shown to be recoverable from Bürmann's theorem of classical analysis. Bosley's technique is then applied to produce a graphical confirmation of the theoretical truncation order in each approximation. The final expressions permit the pressure, temperature, and density to be estimated for any chosen area ratio and gas constant with no intermediate Mach number calculation or tabulation. The techniques are shown in detail so as to facilitate future explorations of transcendental problems where numerical solutions may be difficult to achieve.