Upper panel: emergent spectra in arbitrary units, solution of Eq. (7) for eigenvalues λ_k with k = 1,2,3,4,5 (see Table 1), maximum optical depth τ₀ = 20, black-body temperature kT_bb = 1 keV, and electron temperature kT_e = 10 keV. Thin solid line: energy flux for k = 1. Long-dashed line: k = 2. Dot-dashed line: k = 3. Short-dashed line: k = 4. Dotted line: k = 5. Thick solid line: total emergent spectra obtained from the sum of the first five terms of the series weighted with the corresponding expansion coefficients c_k. The total emergent spectrum has been multiplied by an arbitrary scaling factor in order to be compared with its components. Lower panel: emergent spectra in arbitrary units, solution of Eq. (7), for spectral index α(τ,λ₁) defined in relation (28) and increasing maximum optical depth with kT_bb and kT_e as specified in the upper panel. Solid line: τ₀ = 5. Long-dashed line: τ₀ = 10. Dot-dashed line: τ₀ = 20. Short-dashed line: τ₀ = 40. Dotted line: τ₀ = 70. The values of the spectral indexes and eigenvalues are given in Table 2