Bessel-Hankel

Bessel function and Hankel transformation

中文版

License

The MIT License

Author

Tche LIU, seistche@gmail.com, USTC

Preface

The document contains some MathJax formulas that GitHub can't render properly, so that it looks slightly messy. Fortunately, for Chrome or Firefox, after installing the two add-ons MathJax Plugin for GitHub (GitHub repo.) or github-mathjax-firefox (GitHub repo.) respectively, these formulas will be perfectly show up. And you can have a try 😏.

Methodology

The repository includes some examples for the Hankel transformation, and involves calculations of Bessel functions of the 1st kind and 2nd kind, their derivatives of the 1st order and 2nd order, and their zeros. Next, I will show you some mathematical formulas about calculations of Bessel function's derivatives and zeros, and the Hankel transformation.

Bessel function's derivatives

Represent $J_\nu(x)$ or $Y_\nu(x)$ by $Z_\nu(x)$ in the follows, and there are recurrence relations of Bessel function:

$$ x Z_{\nu - 1}(x) + x Z_{\nu + 1}(x) = 2\nu Z_\nu(x) $$

$$ Z_\nu'(x) = \frac{\nu}{x} Z_\nu(x) - Z_{\nu + 1}(x) $$

so that

$$ Z_{\nu + 2}(x) = \frac{2(\nu + 1)}{x} Z_{\nu + 1}(x) - Z_\nu(x) $$

$$ Z_{\nu + 1}'(x) = \frac{\nu + 1}{x} Z_{\nu + 1}(x) - Z_{\nu + 2}(x) $$

And it's not difficult to reduce the 2nd derivative:

$$ \begin{align*} Z_\nu''(x) & = \frac{d}{dx} \left[ \frac{\nu}{x} Z_\nu(x) - Z_{\nu + 1}(x) \right] \newline & = - \frac{\nu}{x^2} Z_\nu(x) + \frac{\nu}{x} Z_\nu'(x) - Z_{\nu + 1}'(x) \newline & = - \frac{\nu}{x^2} Z_\nu(x) + \frac{\nu}{x} \left[ \frac{\nu}{x} Z_\nu(x) - Z_{\nu + 1}(x) \right] - \left[ \frac{\nu + 1}{x} Z_{\nu + 1}(x) - \frac{2(\nu + 1)}{x} Z_{\nu + 1}(x) + Z_\nu(x) \right] \newline & = \left( - \frac{\nu}{x^2} + \frac{\nu^2}{x^2} - 1 \right) Z_\nu(x) + \left( - \frac{\nu}{x} - \frac{\nu + 1}{x} + \frac{2\nu + 2}{x} \right) Z_{\nu + 1}(x) \newline & = \frac{1}{x^2} \left[ (\nu^2 - \nu - x^2) Z_\nu(x) + x Z_{\nu + 1}(x) \right] \end{align*} $$

and

$$ \begin{align*} Z_\nu''(x) & = \left( \frac{\nu^2}{x^2} - 1 \right) Z_\nu(x) - \frac{1}{x} \left[ \frac{\nu}{x} Z_\nu(x) - Z_{\nu + 1}(x) \right] \newline & = \left( \frac{\nu^2}{x^2} - 1 \right) Z_\nu(x) - \frac{1}{x} Z_\nu'(x) \end{align*} $$

Bessel function's zeros

Fixed point iteration

According to Wikipedia, the fixed point iteration to solve zeros of Bessel function is:

$$ \begin{align*} x_{n + 1} & = x_n - \frac{Z_\nu(x_n)}{Z_\nu'(x_n)} \newline & = x_n - \frac{x_n Z_\nu(x_n)}{\nu Z_\nu(x_n) - x_n Z_{\nu + 1}(x_n)} \end{align*} $$

Halley's method

Based on the above formulas, these results of product are:

$$ Z_\nu(x) Z_\nu'(x) = Z_\nu(x) \left[ \frac{\nu}{x} Z_\nu(x) - Z_{\nu + 1}(x) \right] = \frac{1}{x^2} x Z_\nu(x) \left[ \nu Z_\nu(x) - x Z_{\nu + 1}(x) \right] $$

$$ [Z_\nu'(x)]^2 = \frac{\nu^2}{x^2} Z_\nu(x) - \frac{2\nu}{x} Z_\nu(x) Z_{\nu + 1}(x) + [Z_{\nu + 1}(x)]^2 = \frac{1}{x^2} \left[ \nu^2 Z_\nu(x) - 2\nu x Z_\nu(x) Z_{\nu + 1}(x) + x^2 [Z_{\nu + 1}(x)]^2 \right] $$

$$ Z_\nu(x) Z_\nu''(x) = \frac{1}{x^2} \left\lbrace (\nu^2 - \nu - x^2) [Z_\nu(x)]^2 + x Z_\nu(x) Z_{\nu + 1}(x) \right\rbrace $$

According to Wikipedia, the iteration of Halley's method to solve zeros of Bessel function is:

$$ \begin{align*} x_{n + 1} & = x_n - \frac{2 Z_\nu(x_n) Z_\nu'(x_n) }{ 2 [Z_\nu'(x_n)]^2 - Z_\nu(x_n) Z_\nu''(x_n) } \newline & = x_n - \frac{ 2 x_n Z_\nu(x_n) [ \nu Z_\nu(x_n) - x_n Z_{\nu + 1}(x_n) ] }{ 2 x_n^2 [Z_{\nu + 1}(x_n)]^2 - (4\nu + 1) x_n Z_\nu(x_n) Z_{\nu + 1}(x_n) + (\nu^2 + \nu + x_n^2) [Z_\nu(x_n)]^2} \end{align*} $$

Hankel transformation

Definition

According to Wikipedia, the Hankel transformation of order $\nu$ of a function $f(r)$ is given by:

$$ F_\nu(k) = \int_0^\infty f(r) J_\nu(k r) r dr $$

(Guptasarma and Singh, 1997)

The type of the Hankel transformation of this paper is (equations 2 and 3):

$$ f(r) = \int_0^\infty K(\lambda) J_\nu(r \lambda) d\lambda \approx \frac{1}{r} \sum_{i = 1}^n K(\lambda_i) W_i^\nu $$

where $ \lambda_i = (1/r) \cdot 10^{a + (i - 1) s} $. Besides, values of the shift factor $ a $, the sampling spacing $ s $ and the $ n $-order filter weights $ W_i^{\nu} $ are given in the paper.

The formula is only applicable for the Hankel transformation based on Bessel function of the first class and order 0 or 1.

There are some examples of standard Hankel transformation to verify our programs in this paper:

The equation number	$K(\lambda)$	$J_\nu$	$f(r)$
4	$e^{ - c \lambda}$	$J_0$	$\frac{1}{\sqrt{c^2 + r^2}}$
5	$\lambda e^{ - c \lambda^2}$	$J_0$	$\frac{1}{2c} e^{ - r^2/(4c)}$
6	$\lambda e^{ - c \lambda}$	$J_0$	$\frac{c}{\sqrt{(c^2 + r^2)^3}}$
7	$\lambda e^{ - c \lambda} + \alpha \lambda^2 e^{ - c \lambda^2}$	$J_1$	$\frac{r}{\sqrt{(c^2 + r^2)^3}} + \alpha \frac{r e^{ - r^2/(4c)}}{4 c^2}$
8	$\lambda e^{ - c \lambda}$	$J_1$	$\frac{r}{\sqrt{(c^2 + r^2)^3}}$
9	$\lambda^2 e^{ - c \lambda^2}$	$J_1$	$\frac{r e^{ - r^2/(4c)}}{4 c^2}$
10	$e^{ - c \lambda}$	$J_1$	$\frac{\sqrt{c^2 + r^2} - c}{r \sqrt{c^2 + r^2}}$

(Ogata, 2005)

The approximation of integrals of the Hankel transformation type of this paper is (equation 5.2):

$$ \begin{align*} \int_0^\infty f(x) J_\nu(x) dx & \approx \pi \sum_{k = 1}^{\infty} \omega_{\nu k} f(x) J_\nu(x) \psi'(t) \newline & = \pi \sum_{k = 1}^{\infty} \omega_{\nu k} f \left( \frac{\pi}{h} \psi(h \xi_{\nu k}) \right) J_\nu \left( \frac{\pi}{h} \psi(h \xi_{\nu k}) \right) \psi'(h \xi_{\nu k}) \newline \end{align*}$$

where $ \omega_{\nu k} = Y_{\nu}(\pi \xi_{\nu |k|}) / J_{\nu + 1} (\pi \xi_{\nu |k|}) $, $ x = \pi / h \cdot \psi(t) $, $ \psi(t) = t \tanh(\pi / 2 \cdot \sinh t) $, $ t = h \xi_{\nu k} $, $ h $ is a positive constant, $ \nu $ is a real constant greater than $ -1 $, and $ \xi_{\nu k} $ is the $ |k| $-th zero (counting from the origin) of $ J_{\nu} (\pi x) $ in such a way that

$$ \cdots < \xi_{\nu, -2} < \xi_{\nu, -1} < 0 < \xi_{\nu, 1} < \xi_{\nu, 2} < \cdots; \xi_{\nu, -k} = - \xi_{\nu, k} $$

Set $ x = r \lambda $, so that $ \lambda = x/r $ and $ d\lambda = 1/r \cdot dx $, and $ x = \pi/h \cdot \psi(h \xi_{\nu k}) $, we can reduce:

$$ \begin{align*} \int_0^\infty f(\lambda) J_\nu(r \lambda) \lambda d\lambda & = \int_0^\infty f \left( \frac{x}{r} \right) J_\nu(x) \frac{x}{r} \cdot \frac{1}{r} dx \newline & = \frac{1}{r^2} \int_0^\infty f \left( \frac{x}{r} \right) J_\nu(x) x dx \newline & \approx \frac{1}{r^2} \left[ \pi \sum_{k = 1}^{\infty} \omega_{\nu k} f \left( \frac{x}{r} \right) J_\nu(x) \psi'(h \xi_{\nu k}) \right] x \end{align*} $$

The formula is applicable for the Hankel transformation based on Bessel function of the first class and an arbitrary order.

Implementation

example_Functions

example_Functions is a fortran module including all examples in (Guptasarma and Singh, 1997) for the Hankel transformation.

Hankel_Transform

Hankel_Transform is a fortran module to implement the Hankel transformation based on (Guptasarma and Singh, 1997).

Because the source file has no main program, as an example, you can compile it by $ gfortran -c Hankel_Transform.F90, but CAN'T run it.

Bessel_Functions

Bessel_Function is a fortran module for calculations of Bessel functions of the 1st and 2nd kind, their derivative of the 1st and 2nd order, and zeros of original functions and derivations of the 1st order. The module is mainly modified from mjyzo.f90.

There are two macros for preprocessing in this source file:

• MOREAU for using Moreau's method to calculate Bessel functions of the 1st and 2nd kind and their derivatives of the 1st order; and if not defined, using fortran intrinsic functions to calculate.
• HALLEY for using Halley's method to solve zeros of Bessel function of the 1st and 2nd kind; and if not defined, using the fixed point iteration to solve.

Because the source file has no main program, as an example, you can compile it by $ gfortran -DHALLEY -c Bessel_Function.F90, but CAN'T run it.

Besides, Bessel Zero Solver is an efficient and reliable matlab function to solve zeros of Bessel function of the 1st and 2nd kind based on Halley's method.

Guptasarma_1997

Guptasarma_1997 is a program based on the Hankel_Transform module to implement the methodology of (Guptasarma and Singh, 1997), and includes all examples in this paper by invoking the example_Functions module.

There is a macro for preprocessing in this source file: FAST for using less sample points to faster finish the Hankel transformation; and if not defined, using more sample points to more exactly finish.

For example, you can compile the source file by $ gfortran -DFAST example_Functions.F90 Hankel_Transform.F90 Guptasarma_1997.F90 -o Guptasarma, and run the program by $ ./Guptasarma.

Ogata_2005

Ogata_2005 is a program based on the Bessel_Function module to apply the methodology of (Ogata, 2005) to Hankel transform, and includes all examples in (Guptasarma and Singh, 1997) by invoking the example_Functions module.

For example, you can compile the source file by $ gfortran -DHALLEY example_Functions.F90 Bessel_Function.F90 Ogata_2005.F90 -o Ogata, and run the program by $ ./Ogata.

Examples

References

• Guptasarma and Singh, 1997. New digital linear filters for Hankel J_0 and J_1 transforms. Geophysical Prospecting, 45, 745-762. https://doi.org/10.1046/j.1365-2478.1997.500292.x
• Ogata, 2005. A numerical integration formula based on the Bessel functions. Publ. RIMS, 41, 949-970. https://doi.org/10.2977/prims/1145474602