uations should be compared with the corresponding exact ones of Appendix A4. We shall see that the net of functions (\[Eqn: phieps\]) converges graphically to the multifunction Eq. (\[Eqn: singular\_eigen\]) as $\varepsilon\rightarrow0$.

In the discretized spectral approximation., the singular eigenfunction $\phi(\mu,\nu)$ is replaced by $\phi_{\varepsilon}(\mu,\nu)$, $\varepsilon\rightarrow0$, with the integral in $\nu$ being replaced by an appropriate sum. The solution Eq. (\[Eqn: CaseSolution\_HR\]) of the physically interesting half-space $x\geq0$ problem then reduces to [@Sengupta1988; @Sengupta1995] $$\Phi_{\varepsilon}(x,\mu)=a(\nu_{0})e^{-x/\nu_{0}}\phi(\mu,\nu_{0})+\sum_{i=1}^{N}a(\nu_{i})e^{-x/\nu_{i}}\phi_{\varepsilon}(\mu,\nu_{i})\qquad\mu\in[0,1]\label{Eqn: DiscSpect_HR}$$

where the nodes $\{\nu_{i}\}_{i=1}^{N}$ are chosen suitably. This discretized spectral approximation to Case’s solution has given surprisingly accurate numerical results for a set of properly chosen nodes when compared with exact calculations. Because of its involved nature [@Case1967], the exact calculations are basically numerical which leads to nonlinear integral equations as part of the solution procedure. To appreciate the enormous complexity of the exact treatment of the half-space problem, we recall that the complete set of eigenfunctions $\{\phi(\mu,\nu_{0}),\{\phi(\mu,\nu)\}_{\nu\in[0,1]}\}$ are orthogonal with respect to the half-range weight function $W(\mu)$ of half-range theory, Eq. (\[Eqn: W(mu)\]), that is expressed only in terms of solution of the nonlinear integral equation Eq. (\[Eqn: Omega(-mu)\]). The solution of a half-space problem then evaluates the coefficients $\{ a(\nu_{0}),a(\nu)_{\nu\in[0,1]}\}$ from the appropriate half range (that is $0\leq\mu\leq1$) orthogonality integrals satisfied by the eigenfunctions $\{\phi(\mu,\nu_{0}),\{\phi(\mu,\nu)\}_{\nu\in[0,1]}\}$ with respect to the weight $W(\mu)$, see Appendix A4 for the necessary details of the half-space problem in neutron transport theory.

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