Function trv::maths::sj_transform_symm_biased#

Function Documentation#

void trv::maths::sj_transform_symm_biased(int ell, int i, int N, double *r, double *a, double *k, double *b)#

Perform the (forward) biased symmetric spherical Bessel transform.

The transform is defined here as

\[ b_\ell(k) = (2\pi k)^{-3/2} \int_0^\infty k \mathrm{d}r \, r^{3/2 + i} J_\ell(k r) a_\ell(r) \,. \]

Note

This is the Hankel transform definition used in cosmological wide-angle expansions,

\[ \xi_\ell^{(n)}(r) = 4\pi \mathrm{i}^\ell \int_0^\infty \frac{\mathrm{d}k \, k^2}{(2\pi)^3} (k r)^{-n} j_\ell(kr) P_\ell^{(n)}(k) \,, \]
i.e. \( a(r) \) corresponds to \( P_\ell^{(n)}(k) \) and \( b(k) \) corresponds to \( \mathrm{i}^{-\ell} \xi_\ell^{(n)}(r)\) with \( n \) identified with i.

Parameters

  • ell – [in] Order of the transform.

  • i – [in] Power-law bias index.

  • N – [in] Sample number.

  • r – [in] Pre-transform sample points.

  • a – [in] Pre-transform sample values.

  • k – [out] Post-transform sample points.

  • b – [out] Post-transform sample values.