/* Orion Sky Lawlor, olawlor@acm.org, 9/23/2000 Computes the Ricatti-Bessel function Psi_n(z). This function is used in, for example, various spherical harmonics applications. */ #include #include #include class ricatti_bessel { public: //Compute the R/B function of order n at z // Picks the most accurate function below static double psi(int n,double z) { if (z<1.0+0.3*n) return inf_series(n,z); else return recurse(n,z); } //Compute derivative with respect to z of // the R/B function of order n at z (via finite difference) static double psiPrime(int n,double z) { double dz=1e-11*z; return 0.5*(psi(n,z+dz)-psi(n,z-dz))/dz; } //Compute the R/B function of order n at z // using an infinite series. // Blows up (does not converge) for large z (>20) // Converges to 1e-15 in under 10 iterations for z<1.0 static double inf_series(int n,double z,int *nIt=NULL); //Compute the R/B function of order n at z // using a simple recursion formula. // Blows up for small z (>0.01) and reasonable n static double recurse(int n,double z); //Compute the R/B function of order n at z // using the Gumprecht and Sliepcevich 1951 // finite series identity. [NOT RECOMMENDED] // Blows up for smallish z (>0.1) and reasonable n static double series(int n,double z); private: static double fact(int n) { register double ret=1; while (n>1) { ret*=n; n--; } return ret; } }; //Compute the R/B function of order n at z // using the Gumprecht and Sliepcevich 1951 // finite series identity. double ricatti_bessel::series(int n,double z) { long double sum_s=0,sum_c=0; int m,top=n/2; for (m=0;m<=top;m++) sum_s+=((m&1)?-1:1)*fact(n+2*m)/ ( fact(2*m)*fact(n-2*m)*pow(2*z,2*m) ); top=(int)floor((n-1)/2.0); for (m=0;m<=top;m++) sum_c+=((m&1)?-1:1)*fact(n+2*m+1)/ ( fact(2*m+1)*fact(n-2*m-1)*pow(2*z,2*m+1) ); return sin(z-(n&3)*(M_PI/2)) * sum_s + cos(z-(n&3)*(M_PI/2)) * sum_c; } //Compute the R/B function of order n at z // using an infinite series. double ricatti_bessel::inf_series(int n,double z,int *nIt) { long double sum=0; int m=0; double err=10.0; while ((fabs(err)>1e-15*fabs(sum)) && (m<100000)) { err=((m&1)?-1:1)*fact(n+m)/ ( fact(m)*fact(2*n+2*m+1)) * pow(z,2*m); sum+=err; m++; } if (nIt!=NULL) *nIt=m; return z*pow(2*z,n)*sum; } //Compute the R/B function of order n at z // using a simple recursion formula: // Zn+1(z)=(2(n+1/2)/z)*Zn(z)+Zn-1(z) // Z0(z)=sin(z), Z1(z)=sinz/z - cos(z) double ricatti_bessel::recurse(int n_stop,double z) { register long double Znm=sin(z); if (n_stop==0) return Znm; register long double Zn=Znm/z-cos(z); int n=1; register long double i_z=2.0/z; while (n <@> ******** Program output: ******** <@> Psi0(0.001)=9.999998e-04 1.000003e+00 4.097840e+00 <@> Psi0(0.001)=1.400000e-03 1.000006e+00 4.097840e+00 <@> Psi0(0.002)=1.959999e-03 1.000011e+00 4.097840e+00 <@> Psi0(0.003)=2.743997e-03 9.999868e-01 4.097840e+00 <@> Psi0(0.004)=3.841591e-03 9.999857e-01 4.097840e+00 <@> Psi0(0.005)=5.378214e-03 9.999857e-01 4.097840e+00 <@> Psi0(0.008)=7.529465e-03 9.999754e-01 4.097840e+00 <@> Psi0(0.011)=1.054116e-02 9.999466e-01 4.097840e+00 <@> Psi0(0.015)=1.475735e-02 9.998831e-01 4.097840e+00 <@> Psi0(0.021)=2.065958e-02 9.997832e-01 4.097840e+00 <@> Psi0(0.029)=2.892143e-02 9.995805e-01 4.097840e+00 <@> Psi0(0.040)=4.048458e-02 9.991709e-01 4.097840e+00 <@> Psi0(0.057)=5.666355e-02 9.984023e-01 4.097840e+00 <@> Psi0(0.079)=7.928817e-02 9.968497e-01 4.097840e+00 <@> Psi0(0.111)=1.108915e-01 9.938248e-01 4.097840e+00 <@> Psi0(0.156)=1.549414e-01 9.879158e-01 4.097840e+00 <@> Psi0(0.218)=2.160776e-01 9.763787e-01 4.097840e+00 <@> Psi0(0.305)=3.002106e-01 9.538612e-01 4.097840e+00 <@> Psi0(0.427)=4.140318e-01 9.102641e-01 4.097840e+00 <@> Psi0(0.598)=5.626852e-01 8.266803e-01 4.097840e+00 <@> Psi0(0.837)=7.424248e-01 6.699361e-01 4.097840e+00 <@> Psi0(1.171)=9.212786e-01 3.889008e-01 4.097840e+00 <@> Psi0(1.640)=9.976134e-01 -6.904460e-02 4.097840e+00 <@> Psi0(2.296)=7.484592e-01 -6.631796e-01 4.097840e+00 <@> Psi0(3.214)=-7.254327e-02 -9.973604e-01 4.097840e+00 <@> Psi0(4.500)=-9.775047e-01 -2.109135e-01 4.097840e+00 <@> Psi0(6.300)=1.664534e-02 9.998643e-01 4.097840e+00 <@> Psi0(8.820)=5.687736e-01 -8.224965e-01 4.097840e+00 <@> Psi0(12.348)=-2.169618e-01 9.761773e-01 4.097840e+00 <@> Psi0(17.287)=-9.999682e-01 7.977581e-03 4.097840e+00 <@> Psi0(24.201)=-8.024017e-01 5.967843e-01 4.097840e+00 <@> Psi0(33.882)=6.252984e-01 -7.803908e-01 4.097840e+00 <@> Psi0(47.435)=-3.059324e-01 -9.520595e-01 4.097840e+00 <@> Psi0(66.409)=-4.216685e-01 -9.067502e-01 4.097840e+00 <@> Psi0(92.972)=-9.567326e-01 2.909670e-01 4.097840e+00 <@> Psi1(0.001)=3.333333e-07 6.666679e-04 4.097840e+00 <@> Psi1(0.001)=6.533332e-07 9.333358e-04 4.097840e+00 <@> Psi1(0.002)=1.280533e-06 1.306678e-03 4.097840e+00 <@> Psi1(0.003)=2.509843e-06 1.829324e-03 4.097840e+00 <@> Psi1(0.004)=4.919290e-06 2.561053e-03 4.097840e+00 <@> Psi1(0.005)=9.641794e-06 3.585476e-03 4.097840e+00 <@> Psi1(0.008)=1.889786e-05 5.019649e-03 4.097840e+00 <@> Psi1(0.011)=3.703961e-05 7.027448e-03 4.097840e+00 <@> Psi1(0.015)=7.259686e-05 9.838098e-03 4.097840e+00 <@> Psi1(0.021)=1.422869e-04 1.377276e-02 4.097840e+00 <@> Psi1(0.029)=2.788709e-04 1.928040e-02 4.097840e+00 <@> Psi1(0.040)=5.465430e-04 2.698802e-02 4.097840e+00 <@> Psi1(0.057)=1.071056e-03 3.777196e-02 4.097840e+00 <@> Psi1(0.079)=2.098621e-03 5.284749e-02 4.097840e+00 <@> Psi1(0.111)=4.110810e-03 7.389665e-02 4.097840e+00 <@> Psi1(0.156)=8.047637e-03 1.032105e-01 4.097840e+00 <@> Psi1(0.218)=1.573673e-02 1.438226e-01 4.097840e+00 <@> Psi1(0.305)=3.070357e-02 1.995134e-01 4.097840e+00 <@> Psi1(0.427)=5.964216e-02 2.743161e-01 4.097840e+00 <@> Psi1(0.598)=1.148558e-01 3.705025e-01 4.097840e+00 <@> Psi1(0.837)=2.174140e-01 4.825746e-01 4.097840e+00 <@> Psi1(1.171)=3.976032e-01 5.818371e-01 4.097840e+00 <@> Psi1(1.640)=6.773853e-01 5.845452e-01 4.097840e+00 <@> Psi1(2.296)=9.891851e-01 3.176011e-01 4.097840e+00 <@> Psi1(3.214)=9.747956e-01 -3.758203e-01 4.097840e+00 <@> Psi1(4.500)=-6.315574e-03 -9.760985e-01 4.097840e+00 <@> Psi1(6.300)=-9.972193e-01 1.749388e-01 4.097840e+00 <@> Psi1(8.820)=8.869827e-01 4.682069e-01 4.097840e+00 <@> Psi1(12.348)=-9.937512e-01 -1.364803e-01 4.097840e+00 <@> Psi1(17.287)=-6.582369e-02 -9.961637e-01 4.097840e+00 <@> Psi1(24.201)=-6.299394e-01 -7.763729e-01 4.097840e+00 <@> Psi1(33.882)=7.988409e-01 6.017253e-01 4.097840e+00 <@> Psi1(47.435)=9.456037e-01 -3.258694e-01 4.097840e+00 <@> Psi1(66.409)=9.004005e-01 -4.352271e-01 4.097840e+00 <@> Psi1(92.972)=-3.012593e-01 -9.534866e-01 4.097840e+00 <@> Psi2(0.001)=6.666666e-11 2.000006e-07 4.097840e+00 <@> Psi2(0.001)=1.829333e-10 3.920033e-07 4.097840e+00 <@> Psi2(0.002)=5.019689e-10 7.683258e-07 4.097840e+00 <@> Psi2(0.003)=1.377402e-09 1.505901e-06 4.097840e+00 <@> Psi2(0.004)=3.779590e-09 2.951568e-06 4.097840e+00 <@> Psi2(0.005)=1.037118e-08 5.785084e-06 4.097840e+00 <@> Psi2(0.008)=2.845848e-08 1.133874e-05 4.097840e+00 <@> Psi2(0.011)=7.808975e-08 2.222379e-05 4.097840e+00 <@> Psi2(0.015)=2.142766e-07 4.355762e-05 4.097840e+00 <@> Psi2(0.021)=5.879663e-07 8.537084e-05 4.097840e+00 <@> Psi2(0.029)=1.613332e-06 1.673196e-04 4.097840e+00 <@> Psi2(0.040)=4.426730e-06 3.279133e-04 4.097840e+00 <@> Psi2(0.057)=1.214558e-05 6.425980e-04 4.097840e+00 <@> Psi2(0.079)=3.332013e-05 1.259019e-03 4.097840e+00 <@> Psi2(0.111)=9.139094e-05 2.465889e-03 4.097840e+00 <@> Psi2(0.156)=2.505644e-04 4.826334e-03 4.097840e+00 <@> Psi2(0.218)=6.864081e-04 9.433483e-03 4.097840e+00 <@> Psi2(0.305)=1.877381e-03 1.838930e-02 4.097840e+00 <@> Psi2(0.427)=5.118723e-03 3.566021e-02 4.097840e+00 <@> Psi2(0.598)=1.387061e-02 6.843740e-02 4.097840e+00 <@> Psi2(0.837)=3.713239e-02 1.286538e-01 4.097840e+00 <@> Psi2(1.171)=9.703692e-02 2.319197e-01 4.097840e+00 <@> Psi2(1.640)=2.415832e-01 3.827512e-01 4.097840e+00 <@> Psi2(2.296)=5.441106e-01 5.151906e-01 4.097840e+00 <@> Psi2(3.214)=9.823769e-01 3.635219e-01 4.097840e+00 <@> Psi2(4.500)=9.732942e-01 -4.389014e-01 4.097840e+00 <@> Psi2(6.300)=-4.915244e-01 -8.411781e-01 4.097840e+00 <@> Psi2(8.820)=-2.670707e-01 9.475472e-01 4.097840e+00 <@> Psi2(12.348)=-2.448081e-02 -9.897831e-01 4.097840e+00 <@> Psi2(17.287)=9.885449e-01 -1.801945e-01 4.097840e+00 <@> Psi2(24.201)=7.243147e-01 -6.897969e-01 4.097840e+00 <@> Psi2(33.882)=-5.545670e-01 8.315816e-01 4.097840e+00 <@> Psi2(47.435)=3.657368e-01 9.301893e-01 4.097840e+00 <@> Psi2(66.409)=4.623439e-01 8.864765e-01 4.097840e+00 <@> Psi2(92.972)=9.470116e-01 -3.216293e-01 4.097840e+00 <@> Psi3(0.001)=9.523809e-15 3.809543e-11 4.097840e+00 <@> Psi3(0.001)=3.658666e-14 1.045340e-10 4.097840e+00 <@> Psi3(0.002)=1.405513e-13 2.868427e-10 4.097840e+00 <@> Psi3(0.003)=5.399418e-13 7.870854e-10 4.097840e+00 <@> Psi3(0.004)=2.074240e-12 2.159759e-09 4.097840e+00 <@> Psi3(0.005)=7.968392e-12 5.926399e-09 4.097840e+00 <@> Psi3(0.008)=3.061133e-11 1.626205e-08 4.097840e+00 <@> Psi3(0.011)=1.175961e-10 4.462287e-08 4.097840e+00 <@> Psi3(0.015)=4.517546e-10 1.224429e-07 4.097840e+00 <@> Psi3(0.021)=1.735440e-09 3.359771e-07 4.097840e+00 <@> Psi3(0.029)=6.666716e-09 9.218948e-07 4.097840e+00 <@> Psi3(0.040)=2.560971e-08 2.529492e-06 4.097840e+00 <@> Psi3(0.057)=9.837367e-08 6.940111e-06 4.097840e+00 <@> Psi3(0.079)=3.778475e-07 1.903858e-05 4.097840e+00 <@> Psi3(0.111)=1.451051e-06 5.221537e-05 4.097840e+00 <@> Psi3(0.156)=5.570688e-06 1.431376e-04 4.097840e+00 <@> Psi3(0.218)=2.137274e-05 3.920112e-04 4.097840e+00 <@> Psi3(0.305)=8.189792e-05 1.071594e-03 4.097840e+00 <@> Psi3(0.427)=3.130607e-04 2.918627e-03 4.097840e+00 <@> Psi3(0.598)=1.190991e-03 7.892086e-03 4.097840e+00 <@> Psi3(0.837)=4.488521e-03 2.103850e-02 4.097840e+00 <@> Psi3(1.171)=1.660458e-02 5.451021e-02 4.097840e+00 <@> Psi3(1.640)=5.919476e-02 1.332928e-01 4.097840e+00 <@> Psi3(2.296)=1.957985e-01 2.882595e-01 4.097840e+00 <@> Psi3(3.214)=5.533871e-01 4.658657e-01 4.097840e+00 <@> Psi3(4.500)=1.087783e+00 2.480872e-01 4.097840e+00 <@> Psi3(6.300)=6.071101e-01 -7.806346e-01 4.097840e+00 <@> Psi3(8.820)=-1.038387e+00 8.613270e-02 4.097840e+00 <@> Psi3(12.348)=9.838380e-01 -2.635142e-01 4.097840e+00 <@> Psi3(17.287)=3.517495e-01 9.275041e-01 4.097840e+00 <@> Psi3(24.201)=7.795823e-01 6.276784e-01 4.097840e+00 <@> Psi3(33.882)=-8.806789e-01 -4.765925e-01 4.097840e+00 <@> Psi3(47.435)=-9.070522e-01 4.231059e-01 4.097840e+00 <@> Psi3(66.409)=-8.655900e-01 5.014469e-01 4.097840e+00 <@> Psi3(92.972)=3.521891e-01 9.356418e-01 4.097840e+00 <@> Psi4(0.001)=1.058201e-18 5.291021e-15 4.097840e+00 <@> Psi4(0.001)=5.691259e-18 2.032604e-14 4.097840e+00 <@> Psi4(0.002)=3.060895e-17 7.808490e-14 4.097840e+00 <@> Psi4(0.003)=1.646223e-16 2.999662e-13 4.097840e+00 <@> Psi4(0.004)=8.853778e-16 1.152352e-12 4.097840e+00 <@> Psi4(0.005)=4.761771e-15 4.426892e-12 4.097840e+00 <@> Psi4(0.008)=2.560992e-14 1.700637e-11 4.097840e+00 <@> Psi4(0.011)=1.377359e-13 6.533135e-11 4.097840e+00 <@> Psi4(0.015)=7.407733e-13 2.509728e-10 4.097840e+00 <@> Psi4(0.021)=3.984019e-12 9.641228e-10 4.097840e+00 <@> Psi4(0.029)=2.142661e-11 3.703710e-09 4.097840e+00 <@> Psi4(0.040)=1.152332e-10 1.422734e-08 4.097840e+00 <@> Psi4(0.057)=6.197077e-10 5.465085e-08 4.097840e+00 <@> Psi4(0.079)=3.332469e-09 2.099037e-07 4.097840e+00 <@> Psi4(0.111)=1.791789e-08 8.060525e-07 4.097840e+00 <@> Psi4(0.156)=9.631481e-08 3.094201e-06 4.097840e+00 <@> Psi4(0.218)=5.174573e-07 1.186919e-05 4.097840e+00 <@> Psi4(0.305)=2.777252e-06 4.546434e-05 4.097840e+00 <@> Psi4(0.427)=1.487619e-05 1.736666e-04 4.097840e+00 <@> Psi4(0.598)=7.937288e-05 6.597427e-04 4.097840e+00 <@> Psi4(0.837)=4.202605e-04 2.479356e-03 4.097840e+00 <@> Psi4(1.171)=2.191770e-03 9.119991e-03 4.097840e+00 <@> Psi4(1.640)=1.109313e-02 3.213649e-02 4.097840e+00 <@> Psi4(2.296)=5.287325e-02 1.036789e-01 4.097840e+00 <@> Psi4(3.214)=2.228094e-01 2.761046e-01 4.097840e+00 <@> Psi4(4.500)=7.188572e-01 4.487802e-01 4.097840e+00 <@> Psi4(6.300)=1.166109e+00 -1.332973e-01 4.097840e+00 <@> Psi4(8.820)=-5.570683e-01 -7.857442e-01 4.097840e+00 <@> Psi4(12.348)=5.822271e-01 7.952244e-01 4.097840e+00 <@> Psi4(17.287)=-8.461093e-01 5.475335e-01 4.097840e+00 <@> Psi4(24.201)=-4.988290e-01 8.620289e-01 4.097840e+00 <@> Psi4(33.882)=3.726193e-01 -9.246752e-01 4.097840e+00 <@> Psi4(47.435)=-4.995914e-01 -8.649292e-01 4.097840e+00 <@> Psi4(66.409)=-5.535839e-01 -8.322461e-01 4.097840e+00 <@> Psi4(92.972)=-9.204948e-01 3.917898e-01 4.097840e+00 <@> Psi5(0.001)=9.620009e-23 5.772024e-19 4.097840e+00 <@> Psi5(0.001)=7.243420e-22 3.104340e-18 4.097840e+00 <@> Psi5(0.002)=5.453959e-21 1.669595e-17 4.097840e+00 <@> Psi5(0.003)=4.106577e-20 8.979346e-17 4.097840e+00 <@> Psi5(0.004)=3.092061e-19 4.829320e-16 4.097840e+00 <@> Psi5(0.005)=2.328178e-18 2.597340e-15 4.097840e+00 <@> Psi5(0.008)=1.753008e-17 1.396909e-14 4.097840e+00 <@> Psi5(0.011)=1.319931e-16 7.512898e-14 4.097840e+00 <@> Psi5(0.015)=9.938425e-16 4.040557e-13 4.097840e+00 <@> Psi5(0.021)=7.483113e-15 2.173080e-12 4.097840e+00 <@> Psi5(0.029)=5.634348e-14 1.168718e-11 4.097840e+00 <@> Psi5(0.040)=4.242272e-13 6.285346e-11 4.097840e+00 <@> Psi5(0.057)=3.194040e-12 3.380178e-10 4.097840e+00 <@> Psi5(0.079)=2.404679e-11 1.817637e-09 4.097840e+00 <@> Psi5(0.111)=1.810190e-10 9.772656e-09 4.097840e+00 <@> Psi5(0.156)=1.362368e-09 5.252764e-08 4.097840e+00 <@> Psi5(0.218)=1.024884e-08 2.821715e-07 4.097840e+00 <@> Psi5(0.305)=7.703389e-08 1.514032e-06 4.097840e+00 <@> Psi5(0.427)=5.780403e-07 8.105685e-06 4.097840e+00 <@> Psi5(0.598)=4.323149e-06 4.320399e-05 4.097840e+00 <@> Psi5(0.837)=3.212370e-05 2.282912e-04 4.097840e+00 <@> Psi5(1.171)=2.356720e-04 1.185785e-03 4.097840e+00 <@> Psi5(1.640)=1.685935e-03 5.952727e-03 4.097840e+00 <@> Psi5(2.296)=1.147020e-02 2.789297e-02 4.097840e+00 <@> Psi5(3.214)=7.049602e-02 1.131453e-01 4.097840e+00 <@> Psi5(4.500)=3.499704e-01 3.299899e-01 4.097840e+00 <@> Psi5(6.300)=1.058805e+00 3.257675e-01 4.097840e+00 <@> Psi5(8.820)=4.699351e-01 -8.234812e-01 4.097840e+00 <@> Psi5(12.348)=-5.594629e-01 8.087707e-01 4.097840e+00 <@> Psi5(17.287)=-7.922597e-01 -6.169586e-01 4.097840e+00 <@> Psi5(24.201)=-9.650863e-01 -2.994429e-01 4.097840e+00 <@> Psi5(33.882)=9.796569e-01 2.280518e-01 4.097840e+00 <@> Psi5(47.435)=8.122627e-01 -5.852141e-01 4.097840e+00 <@> Psi5(66.409)=7.905659e-01 -6.131068e-01 4.097840e+00 <@> Psi5(92.972)=-4.412959e-01 -8.967569e-01 4.097840e+00 <@> Psi6(0.001)=7.400007e-27 5.180017e-23 4.097840e+00 <@> Psi6(0.001)=7.800607e-26 3.900331e-22 4.097840e+00 <@> Psi6(0.002)=8.222892e-25 2.936780e-21 4.097840e+00 <@> Psi6(0.003)=8.668038e-24 2.211222e-20 4.097840e+00 <@> Psi6(0.004)=9.137280e-23 1.664950e-19 4.097840e+00 <@> Psi6(0.005)=9.631923e-22 1.253637e-18 4.097840e+00 <@> Psi6(0.008)=1.015334e-20 9.439305e-18 4.097840e+00 <@> Psi6(0.011)=1.070297e-19 7.107346e-17 4.097840e+00 <@> Psi6(0.015)=1.128234e-18 5.351425e-16 4.097840e+00 <@> Psi6(0.021)=1.189302e-17 4.029333e-15 4.097840e+00 <@> Psi6(0.029)=1.253668e-16 3.033874e-14 4.097840e+00 <@> Psi6(0.040)=1.321500e-15 2.284270e-13 4.097840e+00 <@> Psi6(0.057)=1.392966e-14 1.719853e-12 4.097840e+00 <@> Psi6(0.079)=1.468224e-13 1.294787e-11 4.097840e+00 <@> Psi6(0.111)=1.547394e-12 9.746594e-11 4.097840e+00 <@> Psi6(0.156)=1.630518e-11 7.335003e-10 4.097840e+00 <@> Psi6(0.218)=1.717455e-10 5.517451e-09 4.097840e+00 <@> Psi6(0.305)=1.807683e-09 4.146253e-08 4.097840e+00 <@> Psi6(0.427)=1.899878e-08 3.110043e-07 4.097840e+00 <@> Psi6(0.598)=1.991071e-07 2.324193e-06 4.097840e+00 <@> Psi6(0.837)=2.074957e-06 1.724389e-05 4.097840e+00 <@> Psi6(1.171)=2.138636e-05 1.261251e-04 4.097840e+00 <@> Psi6(1.640)=2.156804e-04 8.968059e-04 4.097840e+00 <@> Psi6(2.296)=2.083206e-03 6.025924e-03 4.097840e+00 <@> Psi6(3.214)=1.845009e-02 3.605480e-02 4.097840e+00 <@> Psi6(4.500)=1.366489e-01 1.677666e-01 4.097840e+00 <@> Psi6(6.300)=6.826470e-01 4.086488e-01 4.097840e+00 <@> Psi6(8.820)=1.143171e+00 -3.077540e-01 4.097840e+00 <@> Psi6(12.348)=-1.080628e+00 -3.436227e-02 4.097840e+00 <@> Psi6(17.287)=3.419739e-01 -9.109574e-01 4.097840e+00 <@> Psi6(24.201)=6.017941e-02 -9.800063e-01 4.097840e+00 <@> Psi6(33.882)=-5.456773e-02 9.893266e-01 4.097840e+00 <@> Psi6(47.435)=6.879528e-01 7.252488e-01 4.097840e+00 <@> Psi6(66.409)=6.845339e-01 7.287187e-01 4.097840e+00 <@> Psi6(92.972)=8.682830e-01 -4.973279e-01 4.097840e+00 <@> Psi7(0.001)=4.933338e-31 3.946683e-27 4.097840e+00 <@> Psi7(0.001)=7.280566e-30 4.160345e-26 4.097840e+00 <@> Psi7(0.002)=1.074458e-28 4.385595e-25 4.097840e+00 <@> Psi7(0.003)=1.585673e-27 4.622929e-24 4.097840e+00 <@> Psi7(0.004)=2.340118e-26 4.873199e-23 4.097840e+00 <@> Psi7(0.005)=3.453520e-25 5.137030e-22 4.097840e+00 <@> Psi7(0.008)=5.096663e-24 5.415135e-21 4.097840e+00 <@> Psi7(0.011)=7.521587e-23 5.708273e-20 4.097840e+00 <@> Psi7(0.015)=1.110024e-21 6.017206e-19 4.097840e+00 <@> Psi7(0.021)=1.638151e-20 6.342886e-18 4.097840e+00 <@> Psi7(0.029)=2.417537e-19 6.686215e-17 4.097840e+00 <@> Psi7(0.040)=3.567690e-18 7.047920e-16 4.097840e+00 <@> Psi7(0.057)=5.264914e-17 7.429118e-15 4.097840e+00 <@> Psi7(0.079)=7.769197e-16 7.830332e-14 4.097840e+00 <@> Psi7(0.111)=1.146366e-14 8.252373e-13 4.097840e+00 <@> Psi7(0.156)=1.691204e-13 8.695322e-12 4.097840e+00 <@> Psi7(0.218)=2.494155e-12 9.158259e-11 4.097840e+00 <@> Psi7(0.305)=3.675920e-11 9.637842e-10 4.097840e+00 <@> Psi7(0.427)=5.410654e-10 1.012638e-08 4.097840e+00 <@> Psi7(0.598)=7.943971e-09 1.060604e-07 4.097840e+00 <@> Psi7(0.837)=1.160580e-07 1.103978e-06 4.097840e+00 <@> Psi7(1.171)=1.679142e-06 1.135179e-05 4.097840e+00 <@> Psi7(1.640)=2.383306e-05 1.139469e-04 4.097840e+00 <@> Psi7(2.296)=3.256950e-04 1.090169e-03 4.097840e+00 <@> Psi7(3.214)=4.126339e-03 9.463708e-03 4.097840e+00 <@> Psi7(4.500)=4.480353e-02 6.695248e-02 4.097840e+00 <@> Psi7(6.300)=3.498696e-01 2.938936e-01 4.097840e+00 <@> Psi7(8.820)=1.215056e+00 1.788161e-01 4.097840e+00 <@> Psi7(12.348)=-5.782551e-01 -7.528081e-01 4.097840e+00 <@> Psi7(17.287)=1.049431e+00 -8.297802e-02 4.097840e+00 <@> Psi7(24.201)=9.974121e-01 -2.283112e-01 4.097840e+00 <@> Psi7(33.882)=-1.000594e+00 1.521554e-01 4.097840e+00 <@> Psi7(47.435)=-6.237221e-01 7.800013e-01 4.097840e+00 <@> Psi7(66.409)=-6.565633e-01 7.537410e-01 4.097840e+00 <@> Psi7(92.972)=5.627050e-01 8.259113e-01 4.097840e+00 <@> Psi8(0.001)=2.901964e-35 2.611775e-31 4.097840e+00 <@> Psi8(0.001)=5.995760e-34 3.854441e-30 4.097840e+00 <@> Psi8(0.002)=1.238787e-32 5.688362e-29 4.097840e+00 <@> Psi8(0.003)=2.559463e-31 8.394705e-28 4.097840e+00 <@> Psi8(0.004)=5.288117e-30 1.238882e-26 4.097840e+00 <@> Psi8(0.005)=1.092580e-28 1.828335e-25 4.097840e+00 <@> Psi8(0.008)=2.257383e-27 2.698244e-24 4.097840e+00 <@> Psi8(0.011)=4.663983e-26 3.982032e-23 4.097840e+00 <@> Psi8(0.015)=9.636250e-25 5.876563e-22 4.097840e+00 <@> Psi8(0.021)=1.990939e-23 8.672494e-21 4.097840e+00 <@> Psi8(0.029)=4.113444e-22 1.279869e-19 4.097840e+00 <@> Psi8(0.040)=8.498627e-21 1.888755e-18 4.097840e+00 <@> Psi8(0.057)=1.755833e-19 2.787297e-17 4.097840e+00 <@> Psi8(0.079)=3.627439e-18 4.113023e-16 4.097840e+00 <@> Psi8(0.111)=7.493476e-17 6.068753e-15 4.097840e+00 <@> Psi8(0.156)=1.547748e-15 8.952796e-14 4.097840e+00 <@> Psi8(0.218)=3.195854e-14 1.320263e-12 4.097840e+00 <@> Psi8(0.305)=6.595060e-13 1.945565e-11 4.097840e+00 <@> Psi8(0.427)=1.359411e-11 2.863039e-10 4.097840e+00 <@> Psi8(0.598)=2.795776e-10 4.201507e-09 4.097840e+00 <@> Psi8(0.837)=5.724411e-09 6.132386e-08 4.097840e+00 <@> Psi8(1.171)=1.161937e-07 8.855719e-07 4.097840e+00 <@> Psi8(1.640)=2.318481e-06 1.252261e-05 4.097840e+00 <@> Psi8(2.296)=4.472498e-05 1.698488e-04 4.097840e+00 <@> Psi8(3.214)=8.066719e-04 2.118559e-03 4.097840e+00 <@> Psi8(4.500)=1.270023e-02 2.222470e-02 4.097840e+00 <@> Psi8(6.300)=1.503981e-01 1.588831e-01 4.097840e+00 <@> Psi8(8.820)=9.233057e-01 3.775690e-01 4.097840e+00 <@> Psi8(12.348)=3.781615e-01 -8.232619e-01 4.097840e+00 <@> Psi8(17.287)=5.686359e-01 7.862791e-01 4.097840e+00 <@> Psi8(24.201)=5.580147e-01 8.129558e-01 4.097840e+00 <@> Psi8(33.882)=-3.884080e-01 -9.088914e-01 4.097840e+00 <@> Psi8(47.435)=-8.851884e-01 -4.744360e-01 4.097840e+00 <@> Psi8(66.409)=-8.328344e-01 -5.562351e-01 4.097840e+00 <@> Psi8(92.972)=-7.774970e-01 6.296027e-01 4.097840e+00 <@> Psi9(0.001)=1.527349e-39 1.527352e-35 4.097840e+00 <@> Psi9(0.001)=4.417929e-38 3.155682e-34 4.097840e+00 <@> Psi9(0.002)=1.277906e-36 6.520003e-33 4.097840e+00 <@> Psi9(0.003)=3.696403e-35 1.347079e-31 4.097840e+00 <@> Psi9(0.004)=1.069202e-33 2.783213e-30 4.097840e+00 <@> Psi9(0.005)=3.092715e-32 5.750432e-29 4.097840e+00 <@> Psi9(0.008)=8.945815e-31 1.188101e-27 4.097840e+00 <@> Psi9(0.011)=2.587615e-29 2.454736e-26 4.097840e+00 <@> Psi9(0.015)=7.484779e-28 5.071681e-25 4.097840e+00 <@> Psi9(0.021)=2.164996e-26 1.047855e-23 4.097840e+00 <@> Psi9(0.029)=6.262292e-25 2.164967e-22 4.097840e+00 <@> Psi9(0.040)=1.811362e-23 4.472913e-21 4.097840e+00 <@> Psi9(0.057)=5.239254e-22 9.241215e-20 4.097840e+00 <@> Psi9(0.079)=1.515367e-20 1.909146e-18 4.097840e+00 <@> Psi9(0.111)=4.382639e-19 3.943804e-17 4.097840e+00 <@> Psi9(0.156)=1.267341e-17 8.145552e-16 4.097840e+00 <@> Psi9(0.218)=3.663815e-16 1.681848e-14 4.097840e+00 <@> Psi9(0.305)=1.058627e-14 3.470335e-13 4.097840e+00 <@> Psi9(0.427)=3.055626e-13 7.151877e-12 4.097840e+00 <@> Psi9(0.598)=8.801782e-12 1.470278e-10 4.097840e+00 <@> Psi9(0.837)=2.525234e-10 3.008086e-09 4.097840e+00 <@> Psi9(1.171)=7.188163e-09 6.096388e-08 4.097840e+00 <@> Psi9(1.640)=2.014746e-07 1.212752e-06 4.097840e+00 <@> Psi9(2.296)=5.477490e-06 2.325261e-05 4.097840e+00 <@> Psi9(3.214)=1.401737e-04 4.141731e-04 4.097840e+00 <@> Psi9(4.500)=3.176400e-03 6.347364e-03 4.097840e+00 <@> Psi9(6.300)=5.597742e-02 7.042846e-02 4.097840e+00 <@> Psi9(8.820)=5.646060e-01 3.471629e-01 4.097840e+00 <@> Psi9(12.348)=1.098900e+00 -4.228060e-01 4.097840e+00 <@> Psi9(17.287)=-4.902276e-01 8.238660e-01 4.097840e+00 <@> Psi9(24.201)=-6.054416e-01 7.831658e-01 4.097840e+00 <@> Psi9(33.882)=8.057134e-01 -6.024317e-01 4.097840e+00 <@> Psi9(47.435)=3.064824e-01 -9.433448e-01 4.097840e+00 <@> Psi9(66.409)=4.433657e-01 -8.929215e-01 4.097840e+00 <@> Psi9(92.972)=-7.048706e-01 -7.092591e-01 4.097840e+00 <@> */