BesselFunction

java.lang.Object
- xal.tools.math.BesselFunction

```
public final class BesselFunction
extends java.lang.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).

See Also:

Math, gov.sns.tools.math.ElementaryFunction

Field Summary

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

Constructor Summary

Constructors
Constructor and Description

BesselFunction()

Constructors
Constructor and Description
`BesselFunction()`

Method Summary

All Methods Static Methods Concrete Methods
Modifier and Type	Method and Description
`static double`	`j0(double x)` Zero-Order Spherical Bessel Function of the First Kind
`static double`	`J0(double x)` Compute the zero^th 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 zero^th 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 Detail
  - 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
- Constructor Detail
  - BesselFunction
```
public BesselFunction()
```
- Method Detail
  - 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)

Class BesselFunction

Bessel Function Implementations

References

Field Summary

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Field Detail

ULPS_BRACKET

PI_BY_2

EPS

SMALL_ARG

Constructor Detail

BesselFunction

Method Detail

J0

J1

Jn

Arbitrary Order Bessel Function of First Kind

Y0

Y1

Yn

Arbitrary Order Bessel Function of Second Kind

sinc

Implementation of the sinc Function

j0

Zero-Order Spherical Bessel Function of the First Kind

j1

First-Order Spherical Bessel Function of the First Kind

j2

Second-Order Spherical Bessel Function of the First Kind

j3

Third-Order Spherical Bessel Function of the First Kind

j4

Fourth-Order Spherical Bessel Function of the First Kind