292 lines
9.1 KiB
C
292 lines
9.1 KiB
C
/*
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* This is a dirty implementation of lrhankel_recpart_fill() that calculates
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* the (cylindrical) Hankel transforms of the regularised part of the spherical Hankel
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* functions that are to be summed in the reciprocal space.
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*
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* For now, only the regularisation with κ == 5 && q <= 2 && n <= 5 is implemented
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* by writing down the explicit formula for each q,n pair and k>k0 vs k<k0 case,
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* only with the help of some macros to make the whole thing shorter.
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*
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* N.B. the results for very small k/k0 differ significantly (sometimes even in the first
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* digit for n >= 3, probably due to catastrophic cancellation (hopefully not due
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* to an error in the formula!). On the other hand,
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* these numbers are tiny in their absolute value, so their contribution to the
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* lattice sum should be negligible.
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*/
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#define MAXQM 1
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#define MAXN 5
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#define MAXKAPPA 5
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#include "bessels.h"
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//#include "mdefs.h"
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#include <complex.h>
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#include <string.h>
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#include <assert.h>
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#define SQ(x) ((x)*(x))
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#define P4(x) (((x)*(x))*((x)*(x)))
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/*
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* General form of the κ == 5 transforms. One usually has to put a (-1) factor
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* to some part of the zeroth term of one of the cases k < k0 or k > k0 in order
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* to stay on the correct branch of complex square root...
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*/
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#define KAPPA5SUM(form) (\
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(form(0, 1)) \
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- 5*(form(1, 1)) \
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+10*(form(2, 1)) \
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-10*(form(3, 1)) \
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+ 5*(form(4, 1)) \
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- (form(5, 1)) \
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)
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#define KAPPA5SUMFF(form) (\
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(form(0, (-1))) \
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- 5*(form(1, 1)) \
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+10*(form(2, 1)) \
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-10*(form(3, 1)) \
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+ 5*(form(4, 1)) \
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- (form(5, 1)) \
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)
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/*
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* Prototype for the individual (per q,n) Bessel transform calculating functions.
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* a, b, d, e, ash are recurring pre-calculated intermediate results, see the definition
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* of lrhankel_recpart_fill() below to see their meaning
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*/
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#define LRHANKELDEF(fname) complex double fname(const double c, const double k0, const double k, \
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const complex double *a, const complex double *b, const complex double *d, \
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const complex double *e, const complex double *ash)
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typedef complex double (*lrhankelspec)(const double, const double, const double,
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const complex double *,
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const complex double *,
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const complex double *,
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const complex double *,
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const complex double *);
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// complex double fun(double c, double k0, double k, ccd *a, ccd *b, ccd *d, ccd *e)
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#define FORMK5Q1N0(i,ff) (ff*e[i])
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LRHANKELDEF(fk5q1n0l){
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return (KAPPA5SUMFF(FORMK5Q1N0))/k0;
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}
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LRHANKELDEF(fk5q1n0s){
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return (KAPPA5SUM(FORMK5Q1N0))/k0;
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}
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#undef FORMK5Q1N0
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#define FORMK5Q1N1(i,ff) (-ff*d[i])
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LRHANKELDEF(fk5q1n1l){
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return (KAPPA5SUMFF(FORMK5Q1N1))/(k0*k);
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}
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LRHANKELDEF(fk5q1n1s){
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return (KAPPA5SUM(FORMK5Q1N1))/(k0*k);
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}
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#undef FORMK5Q1N1
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#define FORMK5Q1N2(i,ff) (ff*e[i] - t*a[i] + ff*t*d[i]*a[i])
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LRHANKELDEF(fk5q1n2l){
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double t = 2/(k*k);
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return (KAPPA5SUMFF(FORMK5Q1N2))/k0;
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}
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LRHANKELDEF(fk5q1n2s){
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double t = 2/(k*k);
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return (KAPPA5SUM(FORMK5Q1N2))/k0;
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}
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#undef FORMK5Q1N2
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#define FORMK5Q1N3(i,ff) (-ff*d[i] * (kk3 + 4*a[i]*a[i]))
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LRHANKELDEF(fk5q1n3l){
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double kk3 = 3*k*k;
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return (KAPPA5SUMFF(FORMK5Q1N3))/(k0*k*k*k);
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}
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LRHANKELDEF(fk5q1n3s){
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double kk3 = 3*k*k;
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return (KAPPA5SUM(FORMK5Q1N3))/(k0*k*k*k);
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}
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#undef FORMK5Q1N3
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#define FORMK5Q1N4(i,ff) (ff*e[i] * (kkkk + kk8*a[i]*a[i] + 8*P4(a[i])))
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LRHANKELDEF(fk5q1n4l){
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double kk8 = k*k*8, kkkk = P4(k);
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return (KAPPA5SUMFF(FORMK5Q1N4))/(k0*kkkk);
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}
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LRHANKELDEF(fk5q1n4s){
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double kk8 = k*k*8, kkkk = P4(k);
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return (KAPPA5SUM(FORMK5Q1N4))/(k0*kkkk);
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}
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#undef FORMK5Q1N4
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#define FORMK5Q1N5(i,ff) (d[i]*(kkkk*(-5*ff+b[i])-ff*kk20*a[i]*a[i]-ff*16*P4(a[i])))
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LRHANKELDEF(fk5q1n5l){
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double kk20 = k*k*20, kkkk = P4(k);
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return (KAPPA5SUMFF(FORMK5Q1N5))/(k0*kkkk*k);
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}
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LRHANKELDEF(fk5q1n5s){
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double kk20 = k*k*20, kkkk = P4(k);
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return (KAPPA5SUM(FORMK5Q1N5))/(k0*kkkk*k);
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}
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#undef FORMK5Q1N5
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#define FORMK5Q2N0(i,ff) (-ash[i])
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LRHANKELDEF(fk5q2n0){
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return (KAPPA5SUM(FORMK5Q2N0)) / (k0*k0);
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}
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const lrhankelspec fk5q2n0s = fk5q2n0, fk5q2n0l = fk5q2n0;
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#undef FORMK5Q2N0
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#define FORMK5Q2N1(i,ff) (ff*b[i]*a[i])
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LRHANKELDEF(fk5q2n1l){
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return (KAPPA5SUMFF(FORMK5Q2N1))/(k*k0*k0);
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}
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LRHANKELDEF(fk5q2n1s){
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return (KAPPA5SUM(FORMK5Q2N1))/(k*k0*k0);
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}
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#undef FORMK5Q2N1
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#define FORMK5Q2N2(i,ff) (-ff*b[i]*a[i]*a[i])
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LRHANKELDEF(fk5q2n2l){
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return (KAPPA5SUMFF(FORMK5Q2N2)) / (k*k*k0*k0);
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}
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LRHANKELDEF(fk5q2n2s){
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return (KAPPA5SUM(FORMK5Q2N2)) / (k*k*k0*k0);
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}
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#if 0
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complex double fk5q3n0l(double c, double k0, double k,
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const complex double *a, const complex double *b, const complex double *d, const complex double *e, const complex double *ash) { // FIXME
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return ( /* FIXME */
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- k*b[0] + a[0] * ash[0]
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+ 5 * k*b[1] + a[1] * ash[1]
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-10 * k*b[2] + a[2] * ash[2]
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+10 * k*b[3] + a[3] * ash[3]
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- 5 * k*b[4] + a[4] * ash[4]
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+ k*b[5] + a[5] * ash[5]
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)/(k0*k0*k0);
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}
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#endif
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#define FORMK5Q2N3(i,ff) (ff*a[i]*b[i]*(kk + 4*a[i]*a[i]))
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LRHANKELDEF(fk5q2n3l){
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double kk = k*k;
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return (KAPPA5SUMFF(FORMK5Q2N3))/(3*k0*k0*kk*k);
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}
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LRHANKELDEF(fk5q2n3s){
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double kk = k*k;
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return (KAPPA5SUM(FORMK5Q2N3))/(3*k0*k0*kk*k);
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}
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#undef FORMK5Q2N3
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#define FORMK5Q2N4(i,ff) (-ff*b[i]*a[i]*a[i]*(kk+2*a[i]*a[i]))
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LRHANKELDEF(fk5q2n4l){
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double kk = k*k;
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return (KAPPA5SUMFF(FORMK5Q2N4))/(k0*k0*kk*kk);
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}
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LRHANKELDEF(fk5q2n4s){
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double kk = k*k;
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return (KAPPA5SUM(FORMK5Q2N4))/(k0*k0*kk*kk);
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}
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#undef FORMK5Q2N4
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#define FORMK5Q2N5(i,ff) (ff*a[i]*b[i]*(kkkk+12*kk*(a[i]*a[i])+16*P4(a[i])) )
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LRHANKELDEF(fk5q2n5l){
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double kk = k*k;
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double kkkk = kk * kk;
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return (
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KAPPA5SUMFF(FORMK5Q2N5)
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+16*120*P4(c)*c // Stirling S2(5,5) is no longer zero
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)/(5*k0*k0*kkkk*k);
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}
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LRHANKELDEF(fk5q2n5s){
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double kk = k*k;
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double kkkk = kk * kk;
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return (
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KAPPA5SUM(FORMK5Q2N5)
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+16*120*P4(c)*c // Stirling S2(5,5) is no longer zero
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)/(5*k0*k0*kkkk*k);
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}
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#undef FORMK5Q2N5
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static const lrhankelspec transfuns_f[MAXKAPPA+1][MAXQM+1][MAXN+1] = {
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{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
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{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
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{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
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{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
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{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
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{{fk5q1n0l,fk5q1n1l,fk5q1n2l,fk5q1n3l,fk5q1n4l,fk5q1n5l},{fk5q2n0,fk5q2n1l,fk5q2n2l,fk5q2n3l,fk5q2n4l,fk5q2n5l}}
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};
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static const lrhankelspec transfuns_n[MAXKAPPA+1][MAXQM+1][MAXN+1] = {
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{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
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{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
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{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
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{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
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{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
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{{fk5q1n0s,fk5q1n1s,fk5q1n2s,fk5q1n3s,fk5q1n4s,fk5q1n5s},{fk5q2n0,fk5q2n1s,fk5q2n2s,fk5q2n3s,fk5q2n4s,fk5q2n5s}}
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};
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void lrhankel_recpart_fill(complex double *target,
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size_t maxp /*max. degree of transformed spherical Hankel fun,
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also the max. order of the Hankel transform */,
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size_t lrk_cutoff,
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complex double *hct,
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unsigned kappa, double c, double k0, double k)
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{
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assert(5 == kappa); // Only kappa == 5 implemented so far
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assert(maxp <= 5); // only n <= implemented so far
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// assert(lrk_cutoff <= TODO);
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const lrhankelspec (*funarr)[MAXQM+1][MAXN+1] = (k>k0) ? transfuns_f : transfuns_n;
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memset(target, 0, maxp*(maxp+1)/2*sizeof(complex double));
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complex double a[kappa+1], b[kappa+1], d[kappa+1], e[kappa+1], ash[kappa+1];
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for (size_t sigma = 0; sigma <= kappa; ++sigma) {
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a[sigma] = (sigma * c - I * k0);
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b[sigma] = csqrt(1+k*k/(a[sigma]*a[sigma]));
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d[sigma] = 1/b[sigma];
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e[sigma] = d[sigma] / a[sigma];
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ash[sigma] = casinh(a[sigma]/k);
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}
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for (size_t ql = 0; (ql <= maxp) && (ql < lrk_cutoff); ++ql) // ql is q-1, i.e. corresponds to the hankel term power
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for (size_t deltam = 0; deltam <= maxp; ++deltam){
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complex double result = funarr[kappa][ql][deltam](c,k0,k,a,b,d,e,ash);
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for (size_t p = 0; p <= maxp; ++p)
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trindex_cd(target,p)[deltam] += result * hankelcoeffs_get(hct,p)[ql];
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}
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}
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#ifdef TESTING
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#include <stdio.h>
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int main() {
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double k0 = 0.7;
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double c = 0.1324;
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double kmin = 0.000;
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double kmax = 20;
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double kstep = 0.001;
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size_t kappa = 5;
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for (double k = kmin; k <= kmax; k += kstep) {
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printf("%f ", k);
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complex double a[kappa+1], b[kappa+1], d[kappa+1], e[kappa+1], ash[kappa+1];
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for (size_t sigma = 0; sigma <= kappa; ++sigma) {
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a[sigma] = (sigma * c - I * k0);
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b[sigma] = csqrt(1+k*k/(a[sigma]*a[sigma]));
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d[sigma] = 1/b[sigma];
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e[sigma] = d[sigma] / a[sigma];
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ash[sigma] = casinh(a[sigma]/k);
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}
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for (size_t qm = 0; qm <= MAXQM; ++qm)
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for (size_t n = 0; n <= MAXN; ++n) {
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//if (/*!*/((qm==1)&&(n==0))){ // not skip q==2, n=0 for now
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// complex double fun(double c, double k0, double k, ccd *a, ccd *b, ccd *d, ccd *e)
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complex double result =
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(k < k0 ? transfuns_n : transfuns_f)[kappa][qm][n](c,k0,k,a,b,d,e,ash);
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printf("%.16e %.16e ", creal(result), cimag(result));
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}
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printf("\n");
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}
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return 0;
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}
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#endif
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