Simplified Pressure-Density Relation

Abstract

The result of section I.6 was that the series for Ψ* converges in an angular sector of the half-plane λ < 0. To extend the expansion of Ψ* beyond that domain by analytic continuation is possible but involves complicated computations. What more, the procedure employed above has the disadvantage that the computation of the functions G _n is increasingly complicated as n increases, and that each G _n involves the previous G _m ’s up to m = n − 1. Above, we have started with the isentropic relation, based on the equation of state of a perfect gas, and developed a representation that can only be carried out in practice with a certain approximation. We may now reverse our method⁴. We may begin with an approximate magnitude concerning the physical function ƒ, thus replacing it by another function ${\bar f}$; this leads to a simple mathematical representation for $\bar \Psi *$, instead of the stream-function Ψ*, which function $\bar \Psi *$ * converges absolutely and uniformly at least in the half-plane λ < 0, i. e., in the whole subsonic region. The computation may now be relatively easy, while we may have to justify our choice of ${\bar f}$ by showing that the hypothetical gas thus described is sufficiently similar to a real physical gas. We may begin with

$$ \bar f = C\,\lambda ^{ - 2} ,\,C > 0, $$

((2.1.1))

instead of the function ƒ as defined and calculated in previous sections. Similarly, as in section I.6, we now obtain functions Ḡ _n , such that one has:

$$ \begin{gathered} \bar G'_{n + 1} \, = \,\bar G_n^{\prime \prime } \, + \,C\,\lambda ^{ - 2} \bar G_n , \hfill \\ \bar G_0 = 1,\,\,\bar G_n ( - \infty ) = \,0,\,n\, > \,0, \hfill \\ \end{gathered} $$

((2.1.2))

whence the general formula:

$$ \bar G_n = n!\,\mu _n ( - \lambda )^{ - n} . $$

((2.1.3))