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<title>GNU Scientific Library &ndash; Reference Manual: Coulomb Wave Functions</title>

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<a name="Coulomb-Wave-Functions"></a>
<div class="header">
<p>
Next: <a href="Coulomb-Wave-Function-Normalization-Constant.html#Coulomb-Wave-Function-Normalization-Constant" accesskey="n" rel="next">Coulomb Wave Function Normalization Constant</a>, Previous: <a href="Normalized-Hydrogenic-Bound-States.html#Normalized-Hydrogenic-Bound-States" accesskey="p" rel="previous">Normalized Hydrogenic Bound States</a>, Up: <a href="Coulomb-Functions.html#Coulomb-Functions" accesskey="u" rel="up">Coulomb Functions</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Coulomb-Wave-Functions-1"></a>
<h4 class="subsection">7.7.2 Coulomb Wave Functions</h4>

<p>The Coulomb wave functions <em>F_L(\eta,x)</em>, <em>G_L(\eta,x)</em> are
described in Abramowitz &amp; 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,
</p>
<div class="example">
<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></div>


<dl>
<dt><a name="index-gsl_005fsf_005fcoulomb_005fwave_005fFG_005fe"></a>Function: <em>int</em> <strong>gsl_sf_coulomb_wave_FG_e</strong> <em>(double <var>eta</var>, double <var>x</var>, double <var>L_F</var>, int <var>k</var>, gsl_sf_result * <var>F</var>, gsl_sf_result * <var>Fp</var>, gsl_sf_result * <var>G</var>, gsl_sf_result * <var>Gp</var>, double * <var>exp_F</var>, double * <var>exp_G</var>)</em></dt>
<dd><p>This function computes the Coulomb wave functions <em>F_L(\eta,x)</em>,
<em>G_{L-k}(\eta,x)</em> and their derivatives 
<em>F'_L(\eta,x)</em>, 
<em>G'_{L-k}(\eta,x)</em>
with respect to <em>x</em>.  The parameters are restricted to <em>L,
L-k &gt; -1/2</em>, <em>x &gt; 0</em> and integer <em>k</em>.  Note that <em>L</em>
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></dd></dl>

<dl>
<dt><a name="index-gsl_005fsf_005fcoulomb_005fwave_005fF_005farray"></a>Function: <em>int</em> <strong>gsl_sf_coulomb_wave_F_array</strong> <em>(double <var>L_min</var>, int <var>kmax</var>, double <var>eta</var>, double <var>x</var>, double <var>fc_array</var>[], double * <var>F_exponent</var>)</em></dt>
<dd><p>This function computes the Coulomb wave function <em>F_L(\eta,x)</em> for
<em>L = Lmin \dots Lmin + kmax</em>, storing the results in <var>fc_array</var>.
In the case of overflow the exponent is stored in <var>F_exponent</var>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fsf_005fcoulomb_005fwave_005fFG_005farray"></a>Function: <em>int</em> <strong>gsl_sf_coulomb_wave_FG_array</strong> <em>(double <var>L_min</var>, int <var>kmax</var>, double <var>eta</var>, double <var>x</var>, double <var>fc_array</var>[], double <var>gc_array</var>[], double * <var>F_exponent</var>, double * <var>G_exponent</var>)</em></dt>
<dd><p>This function computes the functions <em>F_L(\eta,x)</em>,
<em>G_L(\eta,x)</em> for <em>L = Lmin \dots Lmin + kmax</em> 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></dd></dl>

<dl>
<dt><a name="index-gsl_005fsf_005fcoulomb_005fwave_005fFGp_005farray"></a>Function: <em>int</em> <strong>gsl_sf_coulomb_wave_FGp_array</strong> <em>(double <var>L_min</var>, int <var>kmax</var>, double <var>eta</var>, double <var>x</var>, double <var>fc_array</var>[], double <var>fcp_array</var>[], double <var>gc_array</var>[], double <var>gcp_array</var>[], double * <var>F_exponent</var>, double * <var>G_exponent</var>)</em></dt>
<dd><p>This function computes the functions <em>F_L(\eta,x)</em>,
<em>G_L(\eta,x)</em> and their derivatives <em>F'_L(\eta,x)</em>,
<em>G'_L(\eta,x)</em> for <em>L = Lmin \dots Lmin + kmax</em> 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></dd></dl>

<dl>
<dt><a name="index-gsl_005fsf_005fcoulomb_005fwave_005fsphF_005farray"></a>Function: <em>int</em> <strong>gsl_sf_coulomb_wave_sphF_array</strong> <em>(double <var>L_min</var>, int <var>kmax</var>, double <var>eta</var>, double <var>x</var>, double <var>fc_array</var>[], double <var>F_exponent</var>[])</em></dt>
<dd><p>This function computes the Coulomb wave function divided by the argument
<em>F_L(\eta, x)/x</em> for <em>L = Lmin \dots Lmin + kmax</em>, 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 <em>\eta \to 0</em>.
</p></dd></dl>

<hr>
<div class="header">
<p>
Next: <a href="Coulomb-Wave-Function-Normalization-Constant.html#Coulomb-Wave-Function-Normalization-Constant" accesskey="n" rel="next">Coulomb Wave Function Normalization Constant</a>, Previous: <a href="Normalized-Hydrogenic-Bound-States.html#Normalized-Hydrogenic-Bound-States" accesskey="p" rel="previous">Normalized Hydrogenic Bound States</a>, Up: <a href="Coulomb-Functions.html#Coulomb-Functions" accesskey="u" rel="up">Coulomb Functions</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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