Bessel Function of the First Kind

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The Bessel functions of the first kind J_n(x) are defined as the solutions to the Bessel differential equation

which are nonsingular at the origin. They are sometimes also called cylinder functions or cylindrical harmonics. The above plot shows J_n(x) for n=0, 1, 2, ..., 5. The notation J_(z,n) was first used by Hansen (1843) and subsequently by Schlömilch (1857) to denote what is now written J_n(2z) (Watson 1966, p. 14). However, Hansen's definition of the function itself in terms of the generating function

is the same as the modern one (Watson 1966, p. 14). Bessel used the notation I_k^h to denote what is now called the Bessel function of the first kind (Cajori 1993, vol. 2, p. 279).

The Bessel function J_n(z) can also be defined by the contour integral

where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).

The Bessel function of the first kind is implemented in the Wolfram Language as BesselJ [nu, z].

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

Plugging into (1) yields

The indicial equation, obtained by setting n=0, is

Since a_0 is defined as the first nonzero term, k^2-m^2=0, so k=+/-m. Now, if k=m,

First, look at the special case m=-1/2, then (11) becomes

so

Now let n=2l, where l=1, 2, ....

which, using the identity 2^ll!(2l-1)!!=(2l)!, gives

Similarly, letting n=2l+1,

which, using the identity 2^ll!(2l+1)!!=(2l+1)!, gives

Plugging back into (◇) with k=m=-1/2 gives

The Bessel functions of order +/-1/2 are therefore defined as

so the general solution for m=+/-1/2 is

Now, consider a general m!=-1/2. Equation (◇) requires

for n=2, 3, ..., so

for n=2, 3, .... Let n=2l+1, where l=1, 2, ..., then

where f(n,m) is the function of l and m obtained by iterating the recursion relationship down to a_1. Now let n=2l, where l=1, 2, ..., so

Plugging back into (◇),

Now define

where the factorials can be generalized to gamma functions for nonintegral m. The above equation then becomes

Returning to equation (◇) and examining the case k=-m,

However, the sign of m is arbitrary, so the solutions must be the same for +m and -m. We are therefore free to replace -m with -|m|, so

and we obtain the same solutions as before, but with m replaced by |m|.

We can relate J_m(x) and J_(-m)(x) (when m is an integer) by writing

Now let l=l^'+m. Then

But l^'!=infty for l^'=-m,...,-1, so the denominator is infinite and the terms on the left are zero. We therefore have

Note that the Bessel differential equation is second-order, so there must be two linearly independent solutions. We have found both only for |m|=1/2. For a general nonintegral order, the independent solutions are J_m and J_(-m). When m is an integer, the general (real) solution is of the form

where J_m is a Bessel function of the first kind, Y_m (a.k.a. N_m) is the Bessel function of the second kind (a.k.a. Neumann function or Weber function), and C_1 and C_2 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 [0,a] according to

where alpha_(num) is the mth zero of Jnu and delta_(mn) is the Kronecker delta (Arfken 1985, p. 592).

Except when 2m is a negative integer,

where Gamma(x) is the gamma function and M_(0,m) is a Whittaker function.

In terms of a confluent hypergeometric function of the first kind, the Bessel function is written

A derivative identity for expressing higher order Bessel functions in terms of J_0(z) is

where T_n(z) is a Chebyshev polynomial of the first kind. Asymptotic forms for the Bessel functions are

for z<<1 and

for z>>|m^2-1/4| (correcting the condition of Abramowitz and Stegun 1972, p. 364).

A derivative identity is

An integral identity is

Some sum identities are

(which follows from the generating function (◇) with t=1),

(Abramowitz and Stegun 1972, p. 363),

(Abramowitz and Stegun 1972, p. 361),

for n>=1 (Abramowitz and Stegun 1972, p. 361),

(Abramowitz and Stegun 1972, p. 361), and the Jacobi-Anger expansion

which can also be written

The Bessel function addition theorem states

Various integrals can be expressed in terms of Bessel functions

which is Bessel's first integral,

for n=1, 2, ...,

for n=1, 2, ...,

for n>-1/2. The Bessel functions are normalized so that

for positive integral (and real) n. Integrals involving J_1(x) include

Ratios of Bessel functions of the first kind have continued fraction

(Wall 1948, p. 349).

The special case of n=0 gives J_0(z) as the series

(Abramowitz and Stegun 1972, p. 360), or the integral

See also

Bessel Function of the Second Kind, Bessel Function Zeros, Debye's Asymptotic Representation, Dixon-Ferrar Formula, Hansen-Bessel Formula, Kapteyn Series, Kneser-Sommerfeld Formula, Mehler's Bessel Function Formula, Modified Bessel Function of the First Kind, Modified Bessel Function of the Second Kind, Nicholson's Formula, Poisson's Bessel Function Formula, Rayleigh Function, 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 Explore this topic in the MathWorld classroom

Related Wolfram sites

http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Bessel Functions J and Y." §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, J_nu(x)" 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.Cajori, F. A History of Mathematical Notations, Vols. 1-2. New York: Dover, 1993.Hansen, P. A. "Ermittelung der absoluten Störungen in Ellipsen von beliebiger Excentricität und Neigung, I." Schriften der Sternwarte Seeberg. Gotha, 1843.Lehmer, D. H. "Arithmetical Periodicities of Bessel Functions." Ann. Math. 33, 143-150, 1932.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 619-622, 1953.Schlömilch, O. X. "Ueber die Bessel'schen Function." Z. für Math. u. Phys. 2, 137-165, 1857.Spanier, J. and Oldham, K. B. "The Bessel Coefficients J_0(x) and J_1(x)" and "The Bessel Function J_nu(x)." Chs. 52-53 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 509-520 and 521-532, 1987.Wall, H. S. Analytic Theory of Continued Fractions. New York: Chelsea, 1948.Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.

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Bessel Function of the First Kind

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Weisstein, Eric W. "Bessel Function of the First Kind." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

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