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8A: Appendix. Integration of the Equations

Numerical integration of equations 8.5.22-24 is straightforward (by Simpson’s rule, for example) except near perineme (\(x = 1\)) and aponeme (\(x = x_2\)), where the integrands become infinite. Near perineme, however, we can substitute \(x = 1 + \xi\) and near aponeme we can substitute \(x = x_2(1 − \xi)\), and we can expand the integrands as power series in \(\xi\) and integrate term by term. I gather here the following results for the intervals \(x = 1\) to \(x = 1 + \epsilon \)and \(x = x_2 − \epsilon\) to \(x = x_2\), where \(\epsilon\) must be chosen to be sufficiently small that \(\epsilon ^4\) is smaller than the precision required. 

\[I_1=\int^{1+\epsilon}_1[v_0^2(1-1/x^2)+2w_0 \ln x - (\ln x)^2]^{-1/2}dx=M(1+\frac{1}{3}A_1\epsilon + \frac{1}{5}B_1\epsilon^2+\frac{1}{7}C_1\epsilon^3+...)\tag{8A.1}\]

\[I_2=\int^{1+\epsilon}_1[v_0^2(1-1/x^2)+2w_0 \ln x - (\ln x)^2]^{-1/2} \ln x \ dx=M(\frac{1}{3}\epsilon+\frac{1}{5}D_1\epsilon^2+\frac{1}{7}E_1\epsilon^3+...)\tag{8A.2}\]

\[I_3=\int^{1+\epsilon}_1[v_0^2(1-1/x^2)+2w_0 \ln x - (\ln x)^2]^{-1/2} x^{-2} dx=M(1+\frac{1}{3}F_1\epsilon+\frac{1}{5}G_1\epsilon^2+\frac{1}{7}H_1\epsilon^3+...)\tag{8A.3}\]

\[I_4=\int^{x_2}_{x_2-\epsilon}[v_0^2(1-1/x^2)+2w_0 \ln x - (\ln x)^2]^{-1/2} dx=N[1+\frac{1}{3}A_2\epsilon/x_2+\frac{1}{5}B_2(\epsilon/x_2)^2+\frac{1}{7}C_2(\epsilon/x_2)^3+...)\tag{8A.4}\]

\[I_5=\int^{x_2}_{x_2-\epsilon}[v_0^2(1-1/x^2)+2w_0 \ln x - (\ln x)^2]^{-1/2} \ln x \ dx= I_4 \ln x_2 - N[\frac{1}{3}\epsilon/x_2 +\frac{1}{5}D_2(\epsilon/x_2)^2+\frac{1}{7}E_2(\epsilon/x_2)^3+...]\tag{8A.5}\]

\[I_6=\int^{x_2}_{x_2-\epsilon}[v_0^2(1-1/x^2)+2w_0 \ln x - (\ln x)^2]^{-1/2} x^{-2} dx= N[1+\frac{1}{3}F_2\epsilon/x_2+\frac{1}{5}G_2(\epsilon/x_2)^2+\frac{1}{7}H_2(\epsilon/x_2)^3+...]/x_2^2\tag{8A.6}\]

The constants are defined as follows.

\[M = \left( \frac{2\epsilon}{v_0^2+w_0}\right)^{1/2}\tag{8A.7}\]

\[N= \left( \frac{2\epsilon x_2}{\ln x_2 - (v_0/x_2)^2-w_0} \right) ^{1/2}\tag{8A.8}\]

\[a_1=- \frac{3v_0^2+w_0+1}{2(v_0^2+w_0)}\tag{8A.9}\]

\[b_1= \frac{4v_0^2+\frac{2}{3}w_0+1}{2(v_0^2+w_0)}\tag{8A.10}\]

\[c_1= - \frac{5v_0^2+\frac{1}{2}w_0+\frac{11}{12}}{2(v_0^2+w_0)}\tag{8A.11}\]

\[a_2 = \frac{3(v_0/x_2)^2+w_0- \ln x_2 + 1}{2 \left( (v_0/x_2)^2+w_0 - \ln x_2 \right)}\tag{8A.12}\]

\[b_2 = \frac{4(v_0/x_2)^2+\frac{2}{3}w_0-\ln x_2 +1}{2 \left( (v_0/x_2)^2+w_0 - \ln x_2 \right)}\tag{8A.13}\]

\[c_2 = \frac{5(v_0/x_2)^2+\frac{1}{2}w_0- \frac{1}{2}\ln x_2 + \frac{11}{12}}{2 \left( (v_0/x_2)^2+w_0 - \ln x_2 \right)}\tag{8A.14}\]

\[A_n = -\frac{1}{2} a_n\tag{8A.15}\]

\[B_n = - \frac{1}{2}b_n + \frac{3}{8} a_n^2\tag{8A.16}\]

\[C_n = - \frac{1}{2}c_n + \frac{3}{4} a_n b_n - \frac{5}{16} a_n^3\tag{8A.17}\]

\[D_n = A_n +\frac{1}{2} (-1)^n\tag{8A.18}\]

\[E_n = B_n + \frac{1}{2}(-1)^n A_n + \frac{1}{3}\tag{8A.19}\]

\[F_n = A_n +2(-1)^n\tag{8A.20}\]

\[G_n = B_n + 2(-1)^n A_n + 3\tag{8A.21}\]

\[H_n = C_n + 2(-1)^n B_n + 3A_n + 4(-1)^n\tag{8A.22}\]

\[\nonumber n= 1,2\]