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tokens: 9 ) and ( 10 ), and then substituting into equations( 4 ) or ( 7 ) allows us to calculate the discrete terms. We now need to calculate the continuous terms. 2.2 Calculating the transition coefficients for the continuous terms The radial wavefunction of the hydrogen for the discrete R nl and the continuum state can be found in Landau Lifshitz ( 1981 ) and Bethe Salpeter ( 1957 ), where is the confluent hypergeometric function. Then, we can derive the transition coefficients required in equations( 4 ) and ( 7 ) from equations( 13 ) and ( 14 ) using with where m j 1 and j 3 in equations( 16 ) and ( 17 ), respectively, and denotes the m th derivative of the term in square brackets. The evaluation of equation( 18 ) for m 1 can be found in the textbook (Bethe Salpeter 1957 ). Evaluating the continuous term in equation( 7 ) evidently requires evaluating equations( 16 ) and ( 17 ). The latter involve summations over j up to ( n f l 1). Since in principle n f , practical computation forces us to assess the number of summation terms required in order to reach a sufficiently accurate convergence. The transition amplitudes are given by equations( 4 ) and ( 7 ) and the overall continuum term is the sum of the Rayleigh and Raman continuum states, and , which are obtained from where as shown in equations( 4 ) and ( 7 ). Fig. 3 illustrates the rapid cross-section convergence by the fourth term in and with f 2, 3, and 4 s i.e. the green dotted curve is the one adopted. Figure 3. Illustration of the KHT absorption cross-section KHT (relative to Thomson scattering) for the first five lines in the Lyman series. The relevant KHT cross-sections are equations ( 57 ) and ( 58 ) and the Thomson cross-section is given in equation( 5 ). The purposes are (i) to illustrate that the continuous terms are of decreasing importance as more continuous terms are added, and (ii) that the same is true down the Lyman series. Hence, we are justified in simplifying (i.e. truncating) the series calculation for continuous terms, as described in Section 2 . To more clearly distinguish between the various curves, the vertical scale cuts off the line centre (shown in later figures). Five curves are illustrated in each panel, in which all discrete terms are included, but the number of continuous terms is increased from zero to four: grey solid line: no continuous terms are included. ( ) is evaluated from equation( 8 ) with . Blue solid line: the most important continuous transition (Rayleigh scattering) is included (i.e. from initial state i 1 s to all continuous states and back to the final state, f 1 s ) such that ( ) is evaluated from equation( 8 ) with and keeping . Red dashed line: the first two continuous transitions are included (from initial state i 1 s to all continuous states, back to f 1 s and 2 s ) such that ( ) is evaluated with and , where is calculated from equation( 19 ) and the remaining for all possible states above 2 s . Yellow dotteddashed line: the first three