The Bessel functions of the first kind are defined as the solutions to the Bessel Differential Equation

(1) |

To solve the differential equation, apply Frobenius Method using a series solution of the form

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

which, using the identity , gives

(13) |

(14) |

(15) |

(16) |

The Bessel Functions of order are therefore defined as

(17) | |||

(18) |

so the general solution for is

(19) |

(20) |

(21) |

(22) | |||

(23) |

for , 3, .... Let , where , 2, ..., then

(24) |

where is the function of and obtained by iterating the recursion relationship down to . Now let , where , 2, ..., so

(25) |

Plugging back into (9),

(26) |

Now define

(27) |

(28) |

(29) |

(30) |

(31) |

(32) |

(33) |

But for , so the Denominator is infinite and the terms on the right are zero. We therefore have

(34) |

Note that the Bessel Differential Equation is second-order, so there must be two linearly independent solutions.
We have found both only for . For a general nonintegral order, the independent solutions are and
. When is an Integer, the general (real) solution is of the form

(35) |

The Bessel functions are Orthogonal in with respect to the weight factor . Except
when is a Negative Integer,

(36) |

In terms of a Confluent Hypergeometric Function of the First Kind, the Bessel function is written

(37) |

(38) |

(39) |

(40) |

(41) |

(42) |

(43) |

(44) |

(45) |

(46) |

(47) |

Roots of the Function are given in the following table.

zero | ||||||

1 | 2.4048 | 3.8317 | 5.1336 | 6.3802 | 7.5883 | 8.7715 |

2 | 5.5201 | 7.0156 | 8.4172 | 9.7610 | 11.0647 | 12.3386 |

3 | 8.6537 | 10.1735 | 11.6198 | 13.0152 | 14.3725 | 15.7002 |

4 | 11.7915 | 13.3237 | 14.7960 | 16.2235 | 17.6160 | 18.9801 |

5 | 14.9309 | 16.4706 | 17.9598 | 19.4094 | 20.8269 | 22.2178 |

Let be the th Root of the Bessel function , then

(48) |

The Roots of its Derivatives are given in the following table.

zero | ||||||

1 | 3.8317 | 1.8412 | 3.0542 | 4.2012 | 5.3175 | 6.4156 |

2 | 7.0156 | 5.3314 | 6.7061 | 8.0152 | 9.2824 | 10.5199 |

3 | 10.1735 | 8.5363 | 9.9695 | 11.3459 | 12.6819 | 13.9872 |

4 | 13.3237 | 11.7060 | 13.1704 | 14.5858 | 15.9641 | 17.3128 |

5 | 16.4706 | 14.8636 | 16.3475 | 17.7887 | 19.1960 | 20.5755 |

Various integrals can be expressed in terms of Bessel functions

(49) | |||

(50) |

which is Bessel's First Integral,

(51) | |||

(52) |

for , 2, ...,

(53) |

(54) |

for . Integrals involving include

(55) |

(56) |

(57) |

**References**

Abramowitz, M. and Stegun, C. A. (Eds.). ``Bessel Functions and .''
§9.1 in *Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, pp. 358-364, 1972.

Arfken, G. ``Bessel Functions of the First Kind, '' and ``Orthogonality.'' §11.1 and 11.2 in
*Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press, pp. 573-591 and 591-596, 1985.

Lehmer, D. H. ``Arithmetical Periodicities of Bessel Functions.'' *Ann. Math.* **33**, 143-150, 1932.

Le Lionnais, F. *Les nombres remarquables.* Paris: Hermann, p. 25, 1983.

Morse, P. M. and Feshbach, H. *Methods of Theoretical Physics, Part I.* New York:
McGraw-Hill, pp. 619-622, 1953.

Spanier, J. and Oldham, K. B. ``The Bessel Coefficients and '' and ``The Bessel Function .''
Chs. 52-53 in *An Atlas of Functions.* Washington, DC: Hemisphere, pp. 509-520 and 521-532, 1987.

© 1996-9

1999-05-26