trv::maths Namespace Reference

Classes
class	HankelTransform
	Perform the (forward biased) Hankel and associated transforms using the FFTLog algorithm. More...
class	SphericalBesselCalculator
	Interpolated spherical Bessel function \( j_\ell(x) \) of the first kind. More...
class	SphericalBesselTransform
class	SphericalHarmonicCalculator
	Reduced spherical harmonics. More...

Functions
std::complex< double >	eval_complex_in_polar (double r, double theta)
	Evaluate a complex number \( r e^{i \theta} \) in the polar form.
double	get_vec3d_magnitude (std::vector< double > vec)
	Return the magnitude of a 3-d vector.
double	get_vec3d_magnitude (double *vec)
	Return the magnitude of a 3-d vector.
std::complex< double >	eval_lanczos_approx_series (std::complex< double > z)
	Evaluate the Lanczos approximation series \( A_g(z) \) for the gamma function.
std::complex< double >	eval_gamma_lanczos (std::complex< double > z)
	Evaluate the gamma function \( \Gamma(z) \) on the complex plane using the Lanczos approximation.
void	get_lngamma_parts (double x, double y, double &lnr, double &theta)
	Get the real and imaginary parts of the log-gamma function \( \ln\Gamma(z = x + \mathrm{i}\,y) \).
std::complex< double >	eval_gamma_lnratio (double mu, std::complex< double > nu)
	Evaluate the logarithm of the ratio of two gamma functions.
double	wigner_3j (int j1, int j2, int j3, int m1, int m2, int m3)
	Calculate Wigner 3-j symbol.

Variables
const std::complex< double >	M_I
	imaginary unit
const double	eps_coupling = 1.e-9
	zero-tolerance for Wigner 3-j coupling coefficients
const double	gconst_lanczos = 7.
	Lanczos approximation constant.

Function Documentation

eval_complex_in_polar()

std::complex< double > trv::maths::eval_complex_in_polar	(	double	r,
		double	theta )

Evaluate a complex number \( r e^{i \theta} \) in the polar form.

Parameters

r	Modulus \( r \).
theta	Argument \( \theta \).

Returns

Value of the complex number.

Definition at line 44 of file maths.cpp.

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get_vec3d_magnitude() [1/2]

double trv::maths::get_vec3d_magnitude

(

std::vector< double >

vec

)

Return the magnitude of a 3-d vector.

Parameters

vec	A 3-d vector.

Returns

Vector magnitude.

Definition at line 53 of file maths.cpp.

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get_vec3d_magnitude() [2/2]

double trv::maths::get_vec3d_magnitude

(

double *

vec

)

Return the magnitude of a 3-d vector.

Parameters

vec	A 3-d vector.

Returns

Vector magnitude.

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Definition at line 57 of file maths.cpp.

eval_lanczos_approx_series()

std::complex< double > trv::maths::eval_lanczos_approx_series

(

std::complex< double >

z

)

Evaluate the Lanczos approximation series \( A_g(z) \) for the gamma function.

The series approximate the gamma function by

\[ \Gamma(z + 1) = \sqrt{2\pi} (z + g + 1/2)^{z + 1/2} \mathrm{e}^{- z - g - 1/2} A_g(z) \]

where \( g \) is the Lanczos constant.

Here \( g = 7 \) is chosen with \( N = 9 \) terms in the series \( A_g(z) \).

Parameters

z	Complex argument.

Returns

Value of \( A_g(z) \).

Definition at line 73 of file maths.cpp.

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eval_gamma_lanczos()

std::complex< double > trv::maths::eval_gamma_lanczos

(

std::complex< double >

z

)

Evaluate the gamma function \( \Gamma(z) \) on the complex plane using the Lanczos approximation.

Here the Lanczos approximation series \( A_g(z) \) uses \( g = 7 \) with \( N = 9 \) terms.

Parameters

z	Complex argument.

Returns

Value of \( \Gamma(z) \).

See also

trv::eval_lanczos_approx_series()

Definition at line 98 of file maths.cpp.

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get_lngamma_parts()

void trv::maths::get_lngamma_parts	(	double	x,
		double	y,
		double &	lnr,
		double &	theta )

Get the real and imaginary parts of the log-gamma function \( \ln\Gamma(z = x + \mathrm{i}\,y) \).

Parameters

[in]	x	Real part of the complex argument.
[in]	y	Imaginary part of the complex argument.
[out]	lnr	Real part of the log-gamma function value.
[out]	theta	Imaginary part of the log-gamma function value.

See also

gsl_sf_lngamma_complex_e() used to evaluate the log-gamma function with complex argument.

Definition at line 122 of file maths.cpp.

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eval_gamma_lnratio()

std::complex< double > trv::maths::eval_gamma_lnratio	(	double	mu,
		std::complex< double >	nu )

Evaluate the logarithm of the ratio of two gamma functions.

The ratio is of the form \( \Gamma\big(\frac{\mu + \nu + 1}{2}\big) \big/ \Gamma\big(\frac{\mu - \nu + 1}{2}\big) \,, \) where \( \mu \) and \( \mathrm{Re}\,\{\nu\} \) are small, while \( \mathrm{Im}\,\{\nu\} \) may be large, in which case the Stirling approximation is used.

Parameters

mu	Real variable.
nu	Complex variable.

Returns

Logarithmic ratio.

Definition at line 130 of file maths.cpp.

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wigner_3j()

double trv::maths::wigner_3j	(	int	j1,
		int	j2,
		int	j3,
		int	m1,
		int	m2,
		int	m3 )

Calculate Wigner 3-j symbol.

Parameters

j1,j2,j3,m1,m2,m3

Wigner 3-j symbol components.

Returns

Value of the Wigner 3-j symbol.

Definition at line 167 of file maths.cpp.

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Variable Documentation

M_I

const std::complex<double> trv::maths::M_I

extern

imaginary unit

eps_coupling

const double trv::maths::eps_coupling = 1.e-9

zero-tolerance for Wigner 3-j coupling coefficients

Definition at line 165 of file maths.cpp.

gconst_lanczos

const double trv::maths::gconst_lanczos = 7.

Lanczos approximation constant.

Definition at line 71 of file maths.cpp.