acoustotreams.special

Special (mathematical) functions.

Special mathematical functions used in acoustotreams. Some functions are imported from scipy.special and treams.special.

Available functions

Spherical waves

ssw_Psi(l, m, kr, theta, phi)	Singular scalar spherical wave \(\Psi\)
ssw_psi(l, m, x, y, z, theta, phi, k)	Far-field amplitude of singular scalar spherical wave \(\psi\)
vsw_L(l, m, kr, theta, phi)	Singular longitudinal vector spherical wave L
ssw_rPsi(l, m, kr, theta, phi)	Regular scalar spherical wave \(\Psi\)
vsw_rL(l, m, kr, theta, phi)	Regular longitudinal vector spherical wave L
tl_ssw(lambda_, mu, l, m, kr, theta, phi)	Singular translation coefficient of scalar and longitudinal vector spherical waves
tl_ssw_r(lambda_, mu, l, m, kr, theta, phi)	Regular translation coefficient of scalar and longitudinal vector spherical waves

Cylindrical waves

scw_Psi(kz, m, krr, phi, z)	Singular scalar cylindrical wave \(\Psi\)
scw_psi(kz, m, x, y, phi, z, krho)	Far-field amplitude of singular scalar cylindrical wave \(\psi\)
vcw_L(kz, m, krr, phi, z, krho, k)	Singular longitudinal vector cylindrical wave L
scw_rPsi(kz, m, krr, phi, z)	Regular scalar cylindrical wave \(\Psi\)
vcw_rL(kz, m, krr, phi, z, krho, k)	Regular longitudinal vector cylindrical wave L
tl_scw(kz1, mu, kz2, m, krr, phi, z)	Singular translation coefficient of scalar and longitudinal vector cylindrical waves
tl_scw_r(kz1, mu, kz2, m, krr, phi, z)	Regular translation coefficient of scalar and longitudinal vector cylindrical waves

Plane waves

spw_Psi(kx, ky, kz, x, y, z)	Scalar plane wave \(\Psi\)
vpw_L(kx, ky, kz, x, y, z)	Longitudinal vector plane wave L

Those functions are just imported from Scipy. So, one only needs to import this package within acoustotreams.

hankel1(v, z[, out])	Hankel function of the first kind.
hankel2(v, z[, out])	Hankel function of the second kind.
jv(v, z[, out])	Bessel function of the first kind of real order and complex argument.
yv(v, z[, out])	Bessel function of the second kind of real order and complex argument.
spherical_jn(n, z[, derivative])	Spherical Bessel function of the first kind or its derivative.
spherical_yn(n, z[, derivative])	Spherical Bessel function of the second kind or its derivative.

The following functions are just imported from treams. So, one only needs to import this package within treams.

Bessel and Hankel functions, with their spherical counterparts, derivatives

hankel1_d hankel2_d jv_d yv_d spherical_hankel1 spherical_hankel2 spherical_hankel1_d spherical_hankel2_d

Those functions just wrap Scipy functions with special optional arguments to be able to analogously access them like their non-spherical counterparts:

spherical_jn_d spherical_yn_d

Scipy functions with enhanced domain

sph_harm lpmv

Integrals for the Ewald summation

incgamma intkambe

Wigner d- and Wigner D-matrix elements

wignersmalld wignerd

Wigner 3j-symbols

wigner3j

Vector spherical harmonics

pi_fun tau_fun vsh_X vsh_Y vsh_Z

Coordinate system transformations

car2cyl car2sph cyl2car cyl2sph sph2car sph2cyl vcar2cyl vcar2sph vcyl2car vcyl2sph vsph2car vsph2cyl car2pol pol2car vcar2pol vpol2car