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## [Bug-gsl] Error in Coulomb Wave Function Routines

 From: scott pratt Subject: [Bug-gsl] Error in Coulomb Wave Function Routines Date: Sat, 28 Jan 2006 12:45:36 -0500

```Dear GSL folks,

I ran into a problem with your routines for calculating F_L and G_L, the
```
partial-wave solutions for the Coulomb problem. The functions are discontinuous, as the functions flip signs for values of x~3.2 and x~5.8 in the example below.
```
I'm working with GSL 1.7

Thanks for everything, - Scott Pratt

#include <cstdio>
#include <cmath>
#include <gsl/gsl_sf.h>

int main(){
int L=2,dummy;
double x,eta=0.1;
double expF,expG;
double *fc,*gc;
fc=new double[L+1];
gc=new double[L+1];

printf("eta=%g\n",eta);
```
printf("x ____ F(L=0) G(L=0) ___ F(L=1) G(L=1) ___ F(L=2) G(L=2) ___\n");
```  for(x=0.1;x<7;x+=0.05){
dummy=gsl_sf_coulomb_wave_FG_array(0,L,eta,x,fc,gc,&expF,&expG);
expF=exp(expF);
expG=exp(expG);
printf("%4.2f   %7.3f %7.3f   %7.3f %7.3f   %7.3f %7.3f\n",
```
x,fc[0]*expF,gc[0]*expG,fc[1]*expF,gc[1]*expG,fc[2]*expF,gc[2] *expG);
```  }
delete [] fc;
delete [] gc;

return 0;
}

____________________ OUTPUT_______________________

x ____ F(L=0)  G(L=0) ___  F(L=1) G(L=1) ___ F(L=2) G(L=2) ___
0.10     0.098   1.007     0.003  10.199     0.000 305.112
0.15     0.147   1.000     0.007   6.838     0.000 135.856
0.20     0.196   0.990     0.013   5.169     0.001  76.625
0.25     0.244   0.979     0.020   4.178     0.001  49.215
0.30     0.292   0.965     0.029   3.524     0.002  34.329
0.35     0.339   0.949     0.040   3.064     0.003  25.356
0.40     0.385   0.931     0.052   2.723     0.004  19.535
0.45     0.430   0.910     0.065   2.463     0.006  15.547
0.50     0.474   0.887     0.080   2.258     0.008  12.696
0.55     0.517   0.863     0.097   2.094     0.011  10.589
0.60     0.559   0.836     0.114   1.959     0.014   8.988
0.65     0.600   0.807     0.133   1.847     0.018   7.745
0.70     0.639   0.776     0.154   1.752     0.022   6.760
0.75     0.676   0.743     0.175   1.671     0.027   5.967
0.80     0.712   0.709     0.198   1.601     0.032   5.320
0.85     0.746   0.673     0.221   1.540     0.038   4.785
0.90     0.778   0.635     0.246   1.485     0.045   4.338
0.95     0.809   0.595     0.272   1.436     0.053   3.962
1.00     0.837   0.554     0.298   1.392     0.061   3.642
1.05     0.863   0.512     0.325   1.351     0.070   3.368
1.10     0.888   0.469     0.353   1.313     0.080   3.132
1.15     0.910   0.424     0.381   1.277     0.091   2.926
1.20     0.930   0.378     0.410   1.243     0.103   2.747
1.25     0.947   0.332     0.440   1.210     0.115   2.590
1.30     0.962   0.284     0.469   1.178     0.128   2.452
1.35     0.975   0.236     0.499   1.146     0.142   2.329
1.40     0.985   0.187     0.529   1.115     0.157   2.219
1.45     0.993   0.138     0.560   1.084     0.173   2.121
1.50     0.999   0.088     0.590   1.053     0.189   2.034
1.55     1.002   0.039     0.620   1.022     0.206   1.954
1.60     1.003  -0.011     0.649   0.990     0.224   1.882
1.65     1.001  -0.061     0.679   0.958     0.243   1.817
1.70     0.997  -0.111     0.707   0.925     0.263   1.757
1.75     0.990  -0.160     0.736   0.891     0.283   1.701
1.80     0.980  -0.209     0.764   0.857     0.304   1.650
1.85     0.969  -0.258     0.791   0.822     0.325   1.603
1.90     0.955  -0.305     0.817   0.786     0.347   1.558
1.95     0.938  -0.352     0.842   0.749     0.370   1.516
2.00     0.920  -0.399     0.867   0.712     0.393   1.477
2.05     0.899  -0.444     0.890   0.673     0.417   1.439
2.10     0.875  -0.488     0.912   0.634     0.442   1.403
2.15     0.850  -0.531     0.933   0.593     0.466   1.368
2.20     0.822  -0.573     0.953   0.552     0.491   1.334
2.25     0.793  -0.613     0.971   0.511     0.517   1.301
2.30     0.761  -0.651     0.988   0.468     0.542   1.269
2.35     0.728  -0.689     1.003   0.425     0.568   1.237
2.40     0.693  -0.724     1.017   0.381     0.594   1.205
2.45     0.656  -0.757     1.029   0.336     0.620   1.174
2.50     0.617  -0.789     1.040   0.291     0.646   1.143
2.55     0.577  -0.819     1.048   0.245     0.672   1.111
2.60     0.536  -0.847     1.055   0.199     0.698   1.079
2.65     0.493  -0.872     1.060   0.153     0.723   1.047
2.70     0.449  -0.896     1.063   0.106     0.749   1.015
2.75     0.404  -0.917     1.065   0.059     0.774   0.982
2.80     0.357  -0.936     1.064   0.012     0.798   0.949
2.85     0.310  -0.952     1.061  -0.035     0.823   0.915
2.90     0.263  -0.967     1.057  -0.082     0.846   0.881
2.95     0.214  -0.978     1.050  -0.129     0.870   0.846
3.00     0.165  -0.988     1.041  -0.175     0.892   0.810
3.05     0.116  -0.995     1.031  -0.221     0.914   0.774
3.10     0.066  -0.999     1.018  -0.267     0.935   0.737
3.15     0.016  -1.001     1.004  -0.312     0.955   0.699
3.20     0.034   1.001    -0.987   0.357    -0.974  -0.661
3.25     0.084   0.998    -0.968   0.401    -0.992  -0.622
3.30     0.133   0.993    -0.948   0.444    -1.009  -0.582
3.35     0.183   0.985    -0.925   0.486    -1.025  -0.542
3.40     0.232   0.974    -0.901   0.528    -1.040  -0.501
3.45     0.280   0.962    -0.875   0.568    -1.054  -0.459
3.50     0.327   0.946    -0.847   0.607    -1.066  -0.417
3.55     0.374   0.929    -0.817   0.644    -1.077  -0.375
3.60     0.420   0.909    -0.786   0.681    -1.086  -0.332
3.65     0.465   0.887    -0.752   0.716    -1.094  -0.288
3.70     0.508   0.863    -0.718   0.749    -1.101  -0.244
3.75     0.551   0.836    -0.682   0.781    -1.106  -0.200
3.80     0.592   0.808    -0.644   0.811    -1.110  -0.155
3.85     0.631   0.777    -0.605   0.839    -1.112  -0.111
3.90     0.669   0.745    -0.564   0.866    -1.112  -0.066
3.95     0.705   0.711    -0.523   0.891    -1.110  -0.021
4.00     0.740   0.674    -0.480   0.914    -1.107   0.025
4.05     0.773   0.637    -0.436   0.935    -1.103   0.070
4.10     0.803   0.597    -0.392   0.953    -1.096   0.114
4.15     0.832   0.557    -0.346   0.970    -1.088   0.159
4.20     0.859   0.514    -0.300   0.985    -1.078   0.204
4.25     0.883   0.471    -0.253   0.997    -1.066   0.248
4.30     0.906   0.426    -0.205   1.007    -1.052   0.292
4.35     0.926   0.381    -0.157   1.015    -1.037   0.335
4.40     0.944   0.334    -0.109   1.021    -1.020   0.378
4.45     0.959   0.286    -0.060   1.024    -1.001   0.420
4.50     0.972   0.238    -0.012   1.026    -0.980   0.461
4.55     0.983   0.189     0.037   1.024    -0.958   0.501
4.60     0.991   0.140     0.086   1.021    -0.934   0.541
4.65     0.997   0.091     0.134   1.015    -0.908   0.580
4.70     1.000   0.041     0.182   1.007    -0.881   0.617
4.75     1.001  -0.009     0.230   0.997    -0.852   0.654
4.80     0.999  -0.059     0.277   0.984    -0.822   0.689
4.85     0.995  -0.109     0.324   0.969    -0.790   0.723
4.90     0.988  -0.158     0.370   0.952    -0.756   0.756
4.95     0.979  -0.208     0.415   0.933    -0.722   0.787
5.00     0.968  -0.256     0.459   0.912    -0.685   0.817
5.05     0.954  -0.304     0.502   0.888    -0.648   0.845
5.10     0.937  -0.351     0.544   0.863    -0.609   0.872
5.15     0.919  -0.398     0.584   0.836    -0.569   0.897
5.20     0.898  -0.443     0.624   0.806    -0.528   0.920
5.25     0.874  -0.487     0.662   0.775    -0.486   0.942
5.30     0.849  -0.530     0.698   0.742    -0.444   0.961
5.35     0.822  -0.572     0.733   0.707    -0.400   0.979
5.40     0.792  -0.612     0.766   0.671    -0.355   0.995
5.45     0.760  -0.651     0.797   0.633    -0.310   1.009
5.50     0.727  -0.688     0.826   0.594    -0.264   1.021
5.55     0.692  -0.723     0.854   0.553    -0.218   1.030
5.60     0.655  -0.757     0.879   0.511    -0.171   1.038
5.65     0.616  -0.789     0.903   0.468    -0.123   1.044
5.70     0.576  -0.818     0.924   0.423    -0.076   1.048
5.75     0.535  -0.846     0.943   0.378    -0.028   1.049
5.80    -0.492   0.872    -0.960  -0.332    -0.019  -1.048
5.85    -0.448   0.895    -0.975  -0.285    -0.067  -1.045
5.90    -0.402   0.916    -0.987  -0.237    -0.114  -1.041
5.95    -0.356   0.935    -0.997  -0.189    -0.162  -1.033
6.00    -0.309   0.952    -1.005  -0.140    -0.209  -1.024
6.05    -0.261   0.966    -1.010  -0.091    -0.255  -1.013
6.10    -0.213   0.978    -1.013  -0.042    -0.301  -0.999
6.15    -0.164   0.987    -1.014   0.007    -0.346  -0.984
6.20    -0.114   0.994    -1.012   0.057    -0.391  -0.966
6.25    -0.064   0.999    -1.008   0.106    -0.435  -0.946
6.30    -0.014   1.001    -1.001   0.155    -0.478  -0.925
6.35    -0.036  -1.000     0.993  -0.203     0.519   0.901
6.40    -0.085  -0.997     0.981  -0.251     0.560   0.876
6.45    -0.135  -0.992     0.968  -0.299     0.600   0.848
6.50    -0.184  -0.984     0.952  -0.345     0.638   0.819
6.55    -0.233  -0.973     0.934  -0.391     0.675   0.788
6.60    -0.282  -0.960     0.913  -0.436     0.710   0.755
6.65    -0.329  -0.945     0.891  -0.480     0.744   0.721
6.70    -0.376  -0.927     0.866  -0.523     0.777   0.685
6.75    -0.422  -0.908     0.839  -0.565     0.807   0.648
6.80    -0.466  -0.885     0.811  -0.605     0.836   0.609
6.85    -0.510  -0.861     0.780  -0.644     0.863   0.569
6.90    -0.552  -0.834     0.748  -0.681     0.889   0.528
6.95    -0.593  -0.806     0.713  -0.717     0.912   0.486
7.00    -0.633  -0.775     0.677  -0.750     0.933   0.442

% ____________________________________________________________________
% Dept. of Physics and Astronomy         http://www.pa.msu.edu/~pratts
% 4208A BioMedical Physical Sci. Bldg.
% Michigan State University                  (517) 355-9200, ext. 2016
% East Lansing, MI  48824-2320                     fax: (517) 353-4500
% ____________________________________________________________________

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