# §33.23 Methods of Computation

## §33.23(i) Methods for the Confluent Hypergeometric Functions

The methods used for computing the Coulomb functions described below are similar to those in §13.29.

## §33.23(ii) Series Solutions

The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii $\rho$ and $r$, respectively, and may be used to compute the regular and irregular solutions. Cancellation errors increase with increases in $\rho$ and $|r|$, and may be estimated by comparing the final sum of the series with the largest partial sum. Use of extended-precision arithmetic increases the radial range that yields accurate results, but eventually other methods must be employed, for example, the asymptotic expansions of §§33.11 and 33.21.

## §33.23(iii) Integration of Defining Differential Equations

When numerical values of the Coulomb functions are available for some radii, their values for other radii may be obtained by direct numerical integration of equations (33.2.1) or (33.14.1), provided that the integration is carried out in a stable direction (§3.7). Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. On the other hand, the irregular solutions of §§33.2(iii) and 33.14(iii) need to be integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§33.11 and 33.21).

## §33.23(iv) Recurrence Relations

In a similar manner to §33.23(iii) the recurrence relations of §§33.4 or 33.17 can be used for a range of values of the integer $\ell$, provided that the recurrence is carried out in a stable direction (§3.6). This implies decreasing $\ell$ for the regular solutions and increasing $\ell$ for the irregular solutions of §§33.2(iii) and 33.14(iii).

## §33.23(v) Continued Fractions

§33.8 supplies continued fractions for $\mathop{F_{\ell}\/}\nolimits'/\mathop{F_{\ell}\/}\nolimits$ and $\mathop{{H^{\pm}_{\ell}}\/}\nolimits'/\mathop{{H^{\pm}_{\ell}}\/}\nolimits$. Combined with the Wronskians (33.2.12), the values of $\mathop{F_{\ell}\/}\nolimits$, $\mathop{G_{\ell}\/}\nolimits$, and their derivatives can be extracted. Inside the turning points, that is, when $\rho<\mathop{\rho_{\mathrm{tp}}\/}\nolimits\!\left(\eta,\ell\right)$, there can be a loss of precision by a factor of approximately $|\mathop{G_{\ell}\/}\nolimits|^{2}$.

## §33.23(vi) Other Numerical Methods

Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24.

Bardin et al. (1972) describes ten different methods for the calculation of $\mathop{F_{\ell}\/}\nolimits$ and $\mathop{G_{\ell}\/}\nolimits$, valid in different regions of the ($\eta,\rho$)-plane.

Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions.

Noble (2004) obtains double-precision accuracy for $\mathop{W_{-\eta,\mu}\/}\nolimits\!\left(2\rho\right)$ for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7).

## §33.23(vii) WKBJ Approximations

WKBJ approximations (§2.7(iii)) for $\rho>\mathop{\rho_{\mathrm{tp}}\/}\nolimits\!\left(\eta,\ell\right)$ are presented in Hull and Breit (1959) and Seaton and Peach (1962: in Eq. (12) $(\rho-c)/c$ should be $(\rho-c)/\rho$). A set of consistent second-order WKBJ formulas is given by Burgess (1963: in Eq. (16) $3\kappa^{2}+2$ should be $3\kappa^{2}c+2$). Seaton (1984) estimates the accuracies of these approximations.

Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for $\mathop{F_{0}\/}\nolimits$ and $\mathop{G_{0}\/}\nolimits$ in the region inside the turning point: $\rho<\mathop{\rho_{\mathrm{tp}}\/}\nolimits\!\left(\eta,\ell\right)$.