Bessel functions are cylindrically symmetric solutions to a second-order differential equation called the Bessel Equation, which can be written as:

```
(z^2)*F''(z) + z*F'(z) - (z^2 + n^2)*F(z)=0;
```

where F' refers to the first derivative of the function F with respect to z, and F'' is the second derivative. Differential equations of this sort represent a variety of physical problems, including nuclear particle scattering, light transmission through an optical fibre, and heat conduction. There are several different types of Bessel functions. The two that most commonly represent the solutions to physical problems are the Bessel functions of the first kind, and the spherical Bessel functions.

- Skill level:
- Moderate

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## Instructions

- 1
Bessel functions of the first kind can be expressed by the infinite series

`J_nu(x)=((x/2)^2)*Sum_[k=0,infinity]{((-1)^k)*([(x^2)/4]^k)/[k!Gamma(k+nu+1)]}`

Alternatively, for integer nu, the Bessel functions can be given by the integral

`J_nu(x)=(1/pi)*Integral_[0, pi] {cos(nu*tau - x*sin(tau))d-tau}`

- 2
As an alternative, numerical values and interpolation formulas for the Bessel functions may be found in many books of mathematical tables, some available online, such as Briggs and Lowan's "Table of the Bessel Functions," sponsored by the National Bureau of Standards and published in 1943 by the Columbia University Press.

There are also polynomial approximations for the Bessel functions, which can be found in texts such as Abramowitz and Stegun's Handbook of Mathematical Functions. For example, J_0(x) for -3<x<3 can be approximated by

`J_0(x)=1 - 2.250*(x/3)^2 + 1.266*(x/3)^4 - .317*(x/3)^6 + .044*(x/3)^8 - .004*(x/3)^10 +...`

- 3
By far the easiest method of obtaining values of the Bessel functions and using them to represent solutions to physical problems is to use the built-in library functions that come with mathematical software packages. Of course such packages as Mathematica, Matlab, Mathcad, and other dedicated mathematics software have Bessel functions built-in, but even more rudimentary packages now often have Bessel functions as part of their library. For example, the "Grapher" utility that comes with Apple's OS X has a built-in Bessel function.

Remember, there is more than one type of Bessel function, so be careful to verify that you're using the built-in function you think you're using.

- 4
Spherical Bessel functions

There is a special class of Bessel functions that represent the solution to problems with spherical symmetry. Specifically, the spherical Bessel functions represent the radial part of the wavefunction for electrons orbiting a nucleus.

`j_0(kr)=sin(kr) j_1(kr)=sin(kr)/(kr)^2 - cos(kr)/kr j_2(kr)=3*sin(kr)/(kr)^3 - 3*cos(kr)/(kr)^2 -sin(kr)/kr`

The spherical Bessel functions may all be generated from j_0(kr) by the relation

`j_p(kr)=(-r/k)^p * ((1/r)(d/dr))^p j_0(kr)`

Like Bessel functions of the first kind, spherical Bessel functions are included in mathematics packages, and may be found in mathematical tables as well.