hankel — Bessel

Hankel's asymptotic expansion of large values of \a x

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Interface

#include <codecogs/maths/special/bessel/hankel.h>

using namespace Maths::Special::Bessel;

double hankel(double x, double v)

This is not a "Hankel function" in the usual sense (the complex Bessel function of the third kind, also known as the Weber function, formed as a linear combination of Bessel functions of the first and second kinds). Rather, it evaluates <em>Hankel's asymptotic expansion</em>, the classical large-x asymptotic technique used to approximate the Bessel functions $J_v(x)$ and $Y_v(x)$.

Writing $\chi = x - \left(\frac{v}{2} + \frac{1}{4}\right)\pi$ and the asymptotic series

$$P(v,x) \sim \sum_{k=0}^{\infty} (-1)^k \frac{(v,2k)}{(2x)^{2k}}, \qquad Q(v,x) \sim \sum_{k=0}^{\infty} (-1)^k \frac{(v,2k+1)}{(2x)^{2k+1}}$$

(1)

where $(v,k) = \dfrac{(4v^2-1^2)(4v^2-3^2)\cdots(4v^2-(2k-1)^2)}{k! \, 8^k}$, this function returns

$$\sqrt{\frac{2}{\pi x}} \left[ P(v,x) \cos(\chi) - Q(v,x) \sin(\chi) \right] \approx J_v(x)$$

(2)

which is used internally by the Bessel J and Y modules for large arguments.

References

Cephes Math Library Release 2.8: June, 2000

Parameters

input argument

order

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