profiley.helpers.hankel

General quadrature method for Hankel transformations. Based on the algorithm provided in H. Ogata, A Numerical Integration Formula Based on the Bessel Functions, Publications of the Research Institute for Mathematical Sciences, vol. 41, no. 4, pp. 949-970, 2005.

Module Contents

Classes

HankelTransform	The basis of the Hankel Transformation algorithm by Ogata 2005.
SphericalHankelTransform	Perform spherical hankel transforms.

class profiley.helpers.hankel.HankelTransform(nu=0, N=200, h=0.05)

Bases: object

The basis of the Hankel Transformation algorithm by Ogata 2005.

This algorithm is used to solve the equation \(\int_0^\infty f(x) J_\nu(x) dx\) where \(J_\nu(x)\) is a Bessel function of the first kind of order \(nu\), and \(f(x)\) is an arbitrary (slowly-decaying) function.

The algorithm is presented in H. Ogata, A Numerical Integration Formula Based on the Bessel Functions, Publications of the Research Institute for Mathematical Sciences, vol. 41, no. 4, pp. 949-970, 2005.

Parameters

nuint or 0.5, optional, default = 0

The order of the bessel function (of the first kind) J_nu(x)

Nint, optional, default = 100

The number of nodes in the calculation. Generally this must increase for a smaller value of the step-size h.

hfloat, optional, default = 0.1

The step-size of the integration.

_psi(t)

_d_psi(t)

_weight()

_roots(N)

_j(x)

_j1(x)

_x(h)

_f(f, x)

transform(f, ret_err=True, ret_cumsum=False)

Perform the transform of the function f

Parameters

fcallable

A function of one variable, representing \(f(x)\)

ret_errboolean, optional, default = True

Whether to return the estimated error

ret_cumsumboolean, optional, default = False

Whether to return the cumulative sum

class profiley.helpers.hankel.SphericalHankelTransform(nu=0, *args, **kwargs)

Bases: HankelTransform

Perform spherical hankel transforms.

Defined as \(\int_0^\infty f(x) j_\nu(x) dx\)

Note: Only does 0th-order transforms currently.

_f(f, x)

_roots(N)