## Bessel Function of the First Kind

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

 (1)

which are nonsingular at the origin. They are sometimes also called Cylinder Functions or Cylindrical Harmonics. The above plot shows for , 2, ..., 5.

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

 (2)

Plugging into (1) yields
 (3)
 (4)
The Indicial Equation, obtained by setting , is
 (5)

Since is defined as the first Nonzero term, , so . Now, if ,

 (6)

 (7)

 (8)

 (9)

First, look at the special case , then (9) becomes
 (10)

so
 (11)

Now let , where , 2, ....
 (12)

which, using the identity , gives
 (13)

Similarly, letting

 (14)

which, using the identity , gives
 (15)

Plugging back into (2) with gives
 (16)

The Bessel Functions of order are therefore defined as
 (17) (18)

so the general solution for is
 (19)

Now, consider a general . Equation (9) requires
 (20)

 (21)

for , 3, ..., so
 (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)

where the factorials can be generalized to Gamma Functions for nonintegral . The above equation then becomes
 (28)

Returning to equation (5) and examining the case ,
 (29)

However, the sign of is arbitrary, so the solutions must be the same for and . We are therefore free to replace with , so
 (30)

and we obtain the same solutions as before, but with replaced by .
 (31)

We can relate and (when is an Integer) by writing
 (32)

Now let . Then

 (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)

where is a Bessel function of the first kind, (a.k.a. ) is the Bessel Function of the Second Kind (a.k.a. Neumann Function or Weber Function), and and are constants. Complex solutions are given by the Hankel Functions (a.k.a. Bessel Functions of the Third Kind).

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

 (36)

where is the Gamma Function and is a Whittaker Function.

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

 (37)

A derivative identity for expressing higher order Bessel functions in terms of is
 (38)

where is a Chebyshev Polynomial of the First Kind. Asymptotic forms for the Bessel functions are
 (39)

for and
 (40)

for . A derivative identity is
 (41)

An integral identity is
 (42)

Some sum identities are
 (43)

 (44)

and the Jacobi-Anger Expansion
 (45)

which can also be written
 (46)

The Bessel function addition theorem states
 (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)

(Le Lionnais 1983).

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)

for , 2, ...,
 (54)

for . Integrals involving include
 (55)

 (56)

 (57)

See also Bessel Function of the Second Kind, Debye's Asymptotic Representation, Dixon-Ferrar Formula, Hansen-Bessel Formula, Kapteyn Series, Kneser-Sommerfeld Formula, Mehler's Bessel Function Formula, Nicholson's Formula, Poisson's Bessel Function Formula, Schläfli's Formula, Schlömilch's Series, Sommerfeld's Formula, Sonine-Schafheitlin Formula, Watson's Formula, Watson-Nicholson Formula, Weber's Discontinuous Integrals, Weber's Formula, Weber-Sonine Formula, Weyrich's Formula

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.