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C..42 VSXC:

See reference [27] for more details.

\begin{dmath} K= F \left( x,z,p_{{3}},q_{{3}},r_{{3}},t_{{3}},u_{{3}},v_{{3}},\a... ...{2}},q _{{2}},r_{{2}},t_{{2}},u_{{2}},v_{{2}},\alpha_{{2}} \right) ,\end{dmath} where \begin{dmath} x= \left( \chi_{\alpha} \right) ^{2}+ \left( \chi_{\beta} \right) ^{2} ,\end{dmath} \begin{dmath} {\it zs}={\frac {\tau_{s}}{ \left( \rho_{s} \right) ^{5/3}}}-{\it cf} ,\end{dmath} \begin{dmath} z={\frac {\tau_{\alpha}}{ \left( \rho_{\alpha} \right) ^{5/3}}}+{\frac {\tau_{\beta}}{ \left( \rho_{\beta} \right) ^{5/3}}}-2\,{\it cf} ,\end{dmath} \begin{dmath} {\it ds}=1-{\frac { \left( \chi_{s} \right) ^{2}}{4\,{\it zs}+4\,{\it cf}}} ,\end{dmath} \begin{dmath} F \left( x,z,p,q,c,d,e,f,\alpha \right) ={\frac {p}{\lambda \left(... ...f{z}^{2}}{ \left( \lambda \left( x,z,\alpha \right) \right) ^{3}}} ,\end{dmath} \begin{dmath} \lambda \left( x,z,\alpha \right) =1+\alpha\, \left( {x}^{2}+z \right) ,\end{dmath} \begin{dmath} {\it cf}=3/5\,{3}^{2/3} \left( {\pi }^{2} \right) ^{2/3} ,\end{dmath} \begin{dmath} \epsilon \left( \alpha,\beta \right) = \left( \alpha+\beta \right)... ...ht) \left( \zeta \left( \alpha,\beta \right) \right) ^{ 4} \right) ,\end{dmath} \begin{dmath} l \left( \alpha,\beta \right) =1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{ \frac {1}{\pi \, \left( \alpha+\beta \right) }}} ,\end{dmath} \begin{dmath} \zeta \left( \alpha,\beta \right) ={\frac {\alpha-\beta}{\alpha+\beta}} ,\end{dmath} \begin{dmath} \omega \left( z \right) ={\frac { \left( 1+z \right) ^{4/3}+ \left( 1-z \right) ^{4/3}-2}{2\,\sqrt [3]{2}-2}} ,\end{dmath} \begin{dmath} e \left( r,t,u,v,w,x,y,p \right) =-2\,t \left( 1+ur \right) \ln ... ...}{t \left( v\sqrt {r}+wr+x{r}^{3/2}+y{r}^{p+1} \right) }} \right) ,\end{dmath} \begin{dmath} c= 1.709921 ,\end{dmath} \begin{dmath} p = [- 0.98, 0.3271, 0.7035] ,\end{dmath} \begin{dmath} q = [- 0.003557,- 0.03229, 0.007695] ,\end{dmath} \begin{dmath} r = [ 0.00625,- 0.02942, 0.05153] ,\end{dmath} \begin{dmath} t = [- 0.00002354, 0.002134, 0.00003394] ,\end{dmath} \begin{dmath} u = [- 0.0001283,- 0.005452,- 0.001269] ,\end{dmath} \begin{dmath} v = [ 0.0003575, 0.01578, 0.001296] ,\end{dmath} \begin{dmath} \alpha = [ 0.001867, 0.005151, 0.00305] ,\end{dmath} \begin{dmath} T = [ 0.031091, 0.015545, 0.016887] ,\end{dmath} \begin{dmath} U = [ 0.21370, 0.20548, 0.11125] ,\end{dmath} \begin{dmath} V = [ 7.5957, 14.1189, 10.357] ,\end{dmath} \begin{dmath} W = [ 3.5876, 6.1977, 3.6231] ,\end{dmath} \begin{dmath} X = [ 1.6382, 3.3662, 0.88026] ,\end{dmath} \begin{dmath} Y = [ 0.49294, 0.62517, 0.49671] \end{dmath} and \begin{dmath} P = [1,1,1] .\end{dmath} To avoid singularities in the limit $\rho_{\bar{s}}\rightarrow 0$ \begin{dmath} G= \left( \rho_{s} \right) ^{4/3}F \left( \chi_{s},{\it zs},p_{{1... ...{2}},q _{{2}},r_{{2}},t_{{2}},u_{{2}},v_{{2}},\alpha_{{2}} \right) .\end{dmath}

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Oct 10, 2007