Package xal.tools.math

Class BesselFunction

java.lang.Object

xal.tools.math.BesselFunction

public final class BesselFunction
extends Object

Bessel Function Implementations

Utility case for computing various Bessel functions that are not included in the Java API.

Implementations for integer-order cylindrical Bessel functions of the first and second kind (i.e., J_n(x) and Y_n(x), n = 1,2,3,... )were taken from www.koders.com and the copyright notice is included in the Java source file. The implementation is based upon that presented in Numerical Recipes by W.H. Press, et. al.

Spherical Bessel functions j_n(x) can be represented as cylindrical Bessel functions with half-integer order. Specifically, we have

j_n(x) = (π/2x)^1/2J_n+½(x).

However, since half-order cylindrical Bessel functions are not included in this class (they are more difficult to implement), an implementation based upon the above formula is not feasible. Instead, spherical Bessel functions j_n(x) for n = 0, 1, 2, 3, 4 are implemented using their trigonometric representations.

NOTES: (CKA)
· There exists a recurrence relationship between Bessel functions of different orders. For example, if B_n(x) is any cylindrical Bessel function of order n, we have the following:

B_n+1(x) = (2n/x)B_n(x) - B_n-1(x).

However, this formula is numerically unstable for Bessel functions of the first kind J_n(x). Thus, it cannot be used to compute the higher order J_n(x) using recursion over lower orders.
· We can apply the above recurrence relation to spherical Bessel functions by expressing them in terms of cylindrical Bessel functions. Letting b_n(x) represent any spherical Bessel function we have

b_n+1(x) = [(2n+1)/x]b_n(x) - b_n-1(x).

Again, this formula is numerically unstable for the Bessel functions of the first kind j_n(x). However, it can be used to determine the representation of each Bessel function in terms of trigonometric functions.

References

[1] www.koders.com
[2]Numerical Recipes, The Art of Scientific Computing, Third Edition, W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery (Cambridge University Press, Cambridge, 2007).

Author:

Christopher K. Allen

See Also:

• Math
• ElementaryFunction

Field Summary

Fields

Modifier and Type	Field	Description
static final double	EPS	smallest tolerance value - not used at the moment
static final double	PI_BY_2	the value PI/2
static final double	SMALL_ARG	Conditional value where polynomial expansions are employed
static final int	ULPS_BRACKET	number of Units in the Last Place (ULPs) used for bracketing approximately equal values

Method Summary

Modifier and Type	Method	Description
static double	j0(double x)	Zero-Order Spherical Bessel Function of the First Kind
static double	J0(double x)	Compute the zeroth order Bessel function of the first kind, J0(x).
static double	j1(double x)	First-Order Spherical Bessel Function of the First Kind
static double	J1(double x)	Compute the first-order Bessel function of the first kind, J1(x).
static double	j2(double x)	Second-Order Spherical Bessel Function of the First Kind
static double	j3(double x)	Third-Order Spherical Bessel Function of the First Kind
static double	j4(double x)	Fourth-Order Spherical Bessel Function of the First Kind
static double	Jn(int n, double x)	Arbitrary Order Bessel Function of First Kind
static double	sinc(double x)	Implementation of the sinc Function
static double	y0(double x)	Compute the zeroth order Bessel function of the second kind, Y0(x).
static double	y1(double x)	Compute the first-order Bessel function of the second kind, Y1(x).
static double	yN(int n, double x)	Arbitrary Order Bessel Function of Second Kind

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Details

ULPS_BRACKET

public static final int ULPS_BRACKET

number of Units in the Last Place (ULPs) used for bracketing approximately equal values

See Also:

• Constant Field Values

PI_BY_2

public static final double PI_BY_2

the value PI/2

See Also:

• Constant Field Values

EPS

public static final double EPS

smallest tolerance value - not used at the moment

See Also:

• Constant Field Values

SMALL_ARG

public static final double SMALL_ARG

Conditional value where polynomial expansions are employed

See Also:

• Constant Field Values

Method Details

J0

public static double J0(double x)

Compute the zero^th order Bessel function of the first kind, J₀(x).

Parameters:

x - a double value

Returns:

the Bessel function of order zero in the argument, J₀(x).

J1

public static double J1(double x)

Compute the first-order Bessel function of the first kind, J₁(x).

Parameters:

x - a double value

Returns:

the Bessel function of order 1 of the argument, J₁(x).

Jn

public static double Jn(int n,
 double x)

Arbitrary Order Bessel Function of First Kind

Computes integer order Bessel function of the first kind, J_n(x).

This implementation relies upon evaluation of the zero and first order Bessel functions J₀(x) and J₁(x)..

Parameters:

n - integer order

x - a double value

Returns:

the Bessel function of order n of the argument, J_n(x).

See Also:

• J0(double)
• J1(double)

y0

public static double y0(double x)

Compute the zero^th order Bessel function of the second kind, Y₀(x).

This implementation relies upon evaluation of the zero order Bessel function Y₀(x).

Parameters:

x - a double value

Returns:

the Bessel function of the second kind, of order 0 of the argument, Y₀(x).

See Also:

• J0(double)

y1

public static double y1(double x)

Compute the first-order Bessel function of the second kind, Y₁(x).

This implementation relies upon evaluation of the first order Bessel function Y₁(x).

Parameters:

x - a double value

Returns:

the Bessel function of the second kind, of order 1 of the argument, Y₁(x).

See Also:

• J1(double)

yN

public static double yN(int n,
 double x)

Arbitrary Order Bessel Function of Second Kind

Computes integer order Bessel function of the second kind, Y_n(x).

This implementation relies upon evaluation of the zero and first order Bessel functions Y₀(x) and Y₁(x).

Parameters:

n - integer order

x - a double value

Returns:

the Bessel function of the second kind, of order n of the argument, Y_n(x).

sinc

public static double sinc(double x)

Implementation of the sinc Function

The sinc function arises frequency in engineering applications, especially communications and signal processing. It is defined as follows:

sinc(x) ≡ sin(x)/x.

The function is not singular at x = 0, which may easily be verified with L'hopital's rule.

To avoid numerical instability, for small values of x we Taylor expand sinc(x) to sixth order about x = 0.

sinc(x) ≈ 1 - x²/6 + x⁴/120 - x⁶/5040 + O(x⁸).

otherwise we return sin(x)/x.

NOTE: (CKA)
· The sinc function is the zero^th order spherical Bessel function j₀.

Parameters:

x - any real number

Returns:

sinc(x) ≡ sin(x)/x

j0

public static double j0(double x)

Zero-Order Spherical Bessel Function of the First Kind

This function is simply an alias for the sinc function, which is the first spherical Bessel function, j₀(x).

Parameters:

x - any real number (double value)

Returns:

spherical Bessel function of the first kind of order 0, j₀(x)

See Also:

• ElementaryFunction.sinc(double)
• sinc(double)

j1

public static double j1(double x)

First-Order Spherical Bessel Function of the First Kind

Direct computation of the first-order spherical Bessel function of the first kind, j₁(x) in terms of trigonometric functions.

To avoid numerical instability, for small values of x we Taylor expand j₁(x) to seventh order about x = 0.

j₁(x) ≈ x/3 - x³/30 + x⁵/840 - x⁷/45360 + O(x⁹).

otherwise we return

j₁(x) = sin(x)/x² - cos(x)/x.

Parameters:

x - any real number (double value)

Returns:

first-order spherical Bessel function of the first kind, j₁(x)

j2

public static double j2(double x)

Second-Order Spherical Bessel Function of the First Kind

Direct computation of the second-order spherical Bessel function of the first kind, j₂(x) in terms of trigonometric functions.

To avoid numerical instability, for small values of x we Taylor expand j₂(x) to eighth order about x = 0.

j₂(x) ≈ x²/15 - x⁴/210 + x⁶/7560 - x⁸/498960 + O(x¹⁰).

otherwise we return

j₂(x) = (3/x - 1)sin(x)/x - 3cos(x)/x².

Parameters:

x - any real number (double value)

Returns:

second-order spherical Bessel function of the first kind, j₂(x)

j3

public static double j3(double x)

Third-Order Spherical Bessel Function of the First Kind

Direct computation of the third-order spherical Bessel function of the first kind, j₃(x) in terms of trigonometric functions.

To avoid numerical instability, for small values of x we Taylor expand j₃(x) to seventh order about x = 0.

j₃(x) ≈ x³/105 + x⁵/1890 - x⁷/83160 + O(x⁹).

otherwise we return

j₃(x) = (15/x³ - 6/x)sin(x)/x - (1 - 15/x²)cos(x)/x.

Parameters:

x - any real number (double value)

Returns:

third-order spherical Bessel function of the first kind, j₃(x)

j4

public static double j4(double x)

Fourth-Order Spherical Bessel Function of the First Kind

Direct computation of the fourth-order spherical Bessel function of the first kind, j₄(x) in terms of trigonometric functions.

To avoid numerical instability, for small values of x we Taylor expand j₄(x) to eighth order about x = 0.

j₂(x) ≈ x⁴/945 - x⁶/20790 + x⁸/1081080 + O(x¹⁰).

otherwise we return

j₄(x) = (1 - 45/x² + 105/x⁴)sin(x)/x + (10/x - 105/x³)cos(x)/x.

Parameters:

x - any real number (double value)

Returns:

fourth-order spherical Bessel function of the first kind, j₄(x)