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<title>Coulomb Wave Functions - GNU Scientific Library -- Reference Manual</title>
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<a name="Coulomb-Wave-Functions"></a>
Next: <a rel="next" accesskey="n" href="Coulomb-Wave-Function-Normalization-Constant.html">Coulomb Wave Function Normalization Constant</a>,
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<h4 class="subsection">7.7.2 Coulomb Wave Functions</h4>
<p>The Coulomb wave functions F_L(\eta,x), G_L(\eta,x) are
described in Abramowitz & Stegun, Chapter 14. Because there can be a
large dynamic range of values for these functions, overflows are handled
gracefully. If an overflow occurs, <code>GSL_EOVRFLW</code> is signalled and
exponent(s) are returned through the modifiable parameters <var>exp_F</var>,
<var>exp_G</var>. The full solution can be reconstructed from the following
relations,
<pre class="example"> F_L(eta,x) = fc[k_L] * exp(exp_F)
G_L(eta,x) = gc[k_L] * exp(exp_G)
F_L'(eta,x) = fcp[k_L] * exp(exp_F)
G_L'(eta,x) = gcp[k_L] * exp(exp_G)
</pre>
<p class="noindent">
<div class="defun">
— Function: int <b>gsl_sf_coulomb_wave_FG_e</b> (<var>double eta, double x, double L_F, int k, gsl_sf_result * F, gsl_sf_result * Fp, gsl_sf_result * G, gsl_sf_result * Gp, double * exp_F, double * exp_G</var>)<var><a name="index-gsl_005fsf_005fcoulomb_005fwave_005fFG_005fe-398"></a></var><br>
<blockquote><p>This function computes the Coulomb wave functions F_L(\eta,x),
<!-- {$G_{L-k}(\eta,x)$} -->
G_{L-k}(\eta,x) and their derivatives
F'_L(\eta,x),
<!-- {$G'_{L-k}(\eta,x)$} -->
G'_{L-k}(\eta,x)
with respect to x. The parameters are restricted to L,
L-k > -1/2, x > 0 and integer k. Note that L
itself is not restricted to being an integer. The results are stored in
the parameters <var>F</var>, <var>G</var> for the function values and <var>Fp</var>,
<var>Gp</var> for the derivative values. If an overflow occurs,
<code>GSL_EOVRFLW</code> is returned and scaling exponents are stored in
the modifiable parameters <var>exp_F</var>, <var>exp_G</var>.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_sf_coulomb_wave_F_array</b> (<var>double L_min, int kmax, double eta, double x, double fc_array</var>[]<var>, double * F_exponent</var>)<var><a name="index-gsl_005fsf_005fcoulomb_005fwave_005fF_005farray-399"></a></var><br>
<blockquote><p>This function computes the Coulomb wave function F_L(\eta,x) for
L = Lmin \dots Lmin + kmax, storing the results in <var>fc_array</var>.
In the case of overflow the exponent is stored in <var>F_exponent</var>.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_sf_coulomb_wave_FG_array</b> (<var>double L_min, int kmax, double eta, double x, double fc_array</var>[]<var>, double gc_array</var>[]<var>, double * F_exponent, double * G_exponent</var>)<var><a name="index-gsl_005fsf_005fcoulomb_005fwave_005fFG_005farray-400"></a></var><br>
<blockquote><p>This function computes the functions F_L(\eta,x),
G_L(\eta,x) for L = Lmin \dots Lmin + kmax storing the
results in <var>fc_array</var> and <var>gc_array</var>. In the case of overflow the
exponents are stored in <var>F_exponent</var> and <var>G_exponent</var>.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_sf_coulomb_wave_FGp_array</b> (<var>double L_min, int kmax, double eta, double x, double fc_array</var>[]<var>, double fcp_array</var>[]<var>, double gc_array</var>[]<var>, double gcp_array</var>[]<var>, double * F_exponent, double * G_exponent</var>)<var><a name="index-gsl_005fsf_005fcoulomb_005fwave_005fFGp_005farray-401"></a></var><br>
<blockquote><p>This function computes the functions F_L(\eta,x),
G_L(\eta,x) and their derivatives F'_L(\eta,x),
G'_L(\eta,x) for L = Lmin \dots Lmin + kmax storing the
results in <var>fc_array</var>, <var>gc_array</var>, <var>fcp_array</var> and <var>gcp_array</var>.
In the case of overflow the exponents are stored in <var>F_exponent</var>
and <var>G_exponent</var>.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_sf_coulomb_wave_sphF_array</b> (<var>double L_min, int kmax, double eta, double x, double fc_array</var>[]<var>, double F_exponent</var>[])<var><a name="index-gsl_005fsf_005fcoulomb_005fwave_005fsphF_005farray-402"></a></var><br>
<blockquote><p>This function computes the Coulomb wave function divided by the argument
F_L(\eta, x)/x for L = Lmin \dots Lmin + kmax, storing the
results in <var>fc_array</var>. In the case of overflow the exponent is
stored in <var>F_exponent</var>. This function reduces to spherical Bessel
functions in the limit \eta \to 0.
</p></blockquote></div>
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