hankel API Summary¶

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.

Classes¶

`HankelTransform`([nu, N, h])	The basis of the Hankel Transformation algorithm by Ogata 2005.
`SymmetricFourierTransform`([ndim, a, b, N, h])	Determine the Fourier Transform of a radially symmetric function in arbitrary dimensions.

Functions¶

get_h(f, nu[, K, cls, hstart, hdecrement, …]) Determine the largest value of h which gives a converged solution.

class hankel.HankelTransform(nu=0, N=None, h=0.05)[source]¶

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.

This class provides a method for directly performing this integration, and also for doing a Hankel Transform.

Parameters:

nu : scalar, optional: The order of the bessel function (of the first kind) J_nu(x)
N : int, optional, default = pi/h: The number of nodes in the calculation. Generally this must increase for a smaller value of the step-size h. Default value is based on where the series will truncate according to the double-exponential convergence to the roots of the Bessel function.
h : float, optional: The step-size of the integration.

classmethod G(f, h, k=None, *args, **kwargs)[source]¶

The absolute value of the non-oscillatory of the summed series’ last term, up to a scaling constant.

This can be used to get the sign of the slope of G with h.

Parameters:	f : callable The function to integrate/transform h : float The resolution parameter of the hankel integration k : float or array-like, optional The scale at which to evaluate the transform. If None, assume an integral.
Returns:	The value of G.

classmethod deltaG(f, h, *args, **kwargs)[source]¶: The slope (up to a constant) of the last term of the series with h

integrate(f, ret_err=True, ret_cumsum=False)[source]¶

Do the Hankel-type integral of the function f.

This is not the Hankel transform, but rather the simplified integral, \(\int_0^\infty f(x) J_\nu(x) dx\) , equivalent to the transform of \(f(r)/r\) at k=1.

Parameters:	f : callable A function of one variable, representing \(f(x)\) ret_err : boolean, optional, default = True Whether to return the estimated error ret_cumsum : boolean, optional, default = False Whether to return the cumulative sum

transform(f, k=1, ret_err=True, ret_cumsum=False, inverse=False)[source]¶

Do the Hankel-transform of the function f.

Parameters:

f : callable: A function of one variable, representing \(f(x)\)
ret_err : boolean, optional, default = True: Whether to return the estimated error
ret_cumsum : boolean, optional, default = False: Whether to return the cumulative sum

Returns:

ret : array-like: The Hankel-transform of f(x) at the provided k. If k is scalar, then this will be scalar.
err : array-like: The estimated error of the approximate integral, at every k. It is merely the last term in the sum. Only returned if ret_err=True.
cumsum : array-like: The total cumulative sum, for which the last term is itself the transform. One can use this to check whether the integral is converging. Only returned if ret_cumsum=True

Notes

The Hankel transform is defined as

\[F(k) = \int_0^\infty r f(r) J_\nu(kr) dr.\]

The inverse transform is identical (swapping k and r of course).

xrange(k=1)[source]¶

Tuple giving (min,max) x value evaluated by f(x).

Parameters:	k : array-like, optional Scales for the transformation. Leave as 1 for an integral.

See also

xrange: the actual x-range under a given choice of parameters.

class hankel.SymmetricFourierTransform(ndim=2, a=1, b=1, N=None, h=0.05)[source]¶

Determine the Fourier Transform of a radially symmetric function in arbitrary dimensions.

Parameters:	ndim : int Number of dimensions the transform is in. a, b : float, default 1 This pair of values defines the Fourier convention used (see Notes below for details) N : int, optional The number of nodes in the calculation. Generally this must increase for a smaller value of the step-size h. h : float, optional The step-size of the integration.

Notes

We allow for arbitrary Fourier convention, according to the scheme in http://mathworld.wolfram.com/FourierTransform.html. That is, we define the forward and inverse n-dimensional transforms respectively as

\[F(k) = \sqrt{\frac{|b|}{(2\pi)^{1-a}}}^n \int f(r) e^{i b\mathbf{k}\cdot\mathbf{r}} d^n\mathbf{r}\]

and

\[f(r) = \sqrt{\frac{|b|}{(2\pi)^{1+a}}}^n \int F(k) e^{-i b\mathbf{k}\cdot\mathbf{r}} d^n \mathbf{k}.\]

By default, we set both a and b to 1, so that the forward transform has a normalisation of unity.

In this general sense, the forward and inverse Hankel transforms are respectively

\[F(k) = \sqrt{\frac{|b|}{(2\pi)^{1-a}}}^n \frac{(2\pi)^{n/2}}{(bk)^{n/2-1}} \int_0^\infty r^{n/2-1} f(r) J_{n/2-1}(bkr) r dr\]

and

\[f(r) = \sqrt{\frac{|b|}{(2\pi)^{1+a}}}^n \frac{(2\pi)^{n/2}}{(br)^{n/2-1}} \int_0^\infty k^{n/2-1} f(k) J_{n/2-1}(bkr) k dk.\]

classmethod G(f, h, k=None, ndim=2)[source]¶

The absolute value of the non-oscillatory part of the summed series’ last term, up to a scaling constant.

This can be used to get the sign of the slope of G with h.

Parameters:	f : callable The function to integrate/transform h : float The resolution parameter of the hankel integration k : float or array-like, optional The scale at which to evaluate the transform. If None, assume an integral. ndim : float The number of dimensions of the transform
Returns:	The value of G.

transform(f, k, *args, **kwargs)[source]¶

Do the n-symmetric Fourier transform of the function f.

Parameters and returns are precisely the same as HankelTransform.transform().

Notes

The n-symmetric fourier transform is defined in terms of the Hankel transform as

\[F(k) = \frac{(2\pi)^{n/2}}{k^{n/2-1}} \int_0^\infty r^{n/2-1} f(r) J_{n/2-1}(kr)r dr.\]

The inverse transform has an inverse normalisation.

classmethod xrange_approx(h, ndim, k=1)[source]¶

Tuple giving approximate (min,max) x value evaluated by f(x/k).

Operates under the assumption that N = pi/h.

Parameters:	h : float The resolution parameter of the Hankel integration ndim : float Number of dimensions of the transform. k : array-like, optional Scales for the transformation. Leave as 1 for an integral.