Bessel Function of the First Kind

The Bessel functions of the first kind $J_n(x)$ are defined as the solutions to the Bessel Differential Equation

$$x^2{d^2y\over dx^2} + x {dy\over dx} + (x^2-m^2)y = 0$$ (1)

which are nonsingular at the origin. They are sometimes also called Cylinder Functions or Cylindrical Harmonics. The above plot shows $J_n(x)$ for $n=1$, 2, ..., 5.

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

$$y = x^k \sum_{n=0}^\infty a_n x^n = \sum_{n=0}^\infty a_nx^{n+k}.$$ (2)

Plugging into (1) yields

$x^2\sum_{n=0}^\infty (k+n)(k+n-1)a_nx^{k+n-2}$
$+ x \sum_{n=0}^\infty (k+n)a_nx^{k+n-1}+x^2\sum_{n=0}^\infty a_nx^{k+n}$
$- m^2 \sum_{n=0}^\infty a_nx^{n+k} = 0\quad$	(3)

$\sum_{n=0}^\infty (k+n)(k+n-1)a_nx^{k+n}+ \sum_{n=0}^\infty (k+n)a_nx^{k+n}$
${}+ \sum_{n=2}^\infty a_{n-2}x^{k+n} -m^2 \sum_{n=0}^\infty a_nx^{n+k} = 0.\qquad\llap{}$	(4)

The Indicial Equation, obtained by setting $n = 0$, is

$$a_0[k(k-1)+k-m^2] = a_0(k^2-m^2) = 0.$$ (5)

Since $a_0$ is defined as the first Nonzero term, $k^2-m^2 = 0$, so $k = \pm m$. Now, if $k = m$,

$$\sum_{n=0}^\infty [(m+n)(m+n-1)+(m+n)-m^2]a_nx^{m+n} + \sum_{n=2}^\infty a_{n-2}x^{m+n} = 0$$ (6)

$$\sum_{n=0}^\infty [(m+n)^2-m^2]a_nx^{m+n} + \sum_{n=2}^\infty a_{n-2}x^{m+n} = 0$$ (7)

$$\sum_{n=0}^\infty n(2m+n)a_nx^{m+n} + \sum_{n=2}^\infty a_{n-2}x^{m+n} = 0$$ (8)

$$a_1(2m+1) + \sum_{n=2}^\infty [a_nn(2m+n)+a_{n-2}]x^{m+n} = 0.$$ (9)

First, look at the special case $m = -{1/2}$, then (9) becomes

$$\sum_{n=2}^\infty [a_nn(n-1)+a_{n-2}]x^{m+n} = 0,$$ (10)

so

$$a_n = - {1\over n(n-1)} a_{n-2}.$$ (11)

Now let $n \equiv 2l$, where $l = 1$, 2, ....

$$a_{2l} = - {1\over 2l(2l-1)} a_{2l-2}$$

$$= {(-1)^l\over[2l(2l-1)][2(l-1)(2l-3)]\cdots[2\cdot 1\cdot 1]} a_0$$

$$= {(-1)^l\over 2^ll!(2l-1)!!} a_0,$$ (12)

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

$$a_{2l} = {(-1)^l\over (2l)!} a_0.$$ (13)

Similarly, letting $n\equiv 2l+1$

$$a_{2l+1} &= -{1\over (2l+1)(2l)} a_{2l-1} = {(-1)^l\over [2l(2l+1)][2(l-1)(2l-1)]\cdots [2\cdot 1\cdot 3][1]} a_1,$$ (14)

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

$$a_{2l+1} = {(-1)^l\over 2^ll!(2l+1)!!} a_1 = {(-1)^l\over (2l+1)!} a_1.$$ (15)

Plugging back into (2) with $k = m = -{1/2}$ gives

$$y = x^{-1/2}\sum_{n=0}^\infty a_nx^n$$

$$= x^{-1/2}\left[{\,\sum_{n=1,3,5,\ldots}^\infty a_nx^n+\sum_{n=0,2,4,\ldots}^\infty a_nx^n}\right]$$

$$= x^{-1/2}\left[{\,\sum_{l=0}^\infty a_{2l}x^{2l}+\sum_{l=0}^\infty a_{2l+1}x^{2l+1}}\right]$$

$$= x^{-1/2}\left[{a_0 \sum_{l=0}^\infty {(-1)^l\over(2l)!} x^{2l}+a_1\sum_{l=0}^\infty {(-1)^l\over(2l+1)!}x^{2l+1}}\right]$$

$$= x^{-1/2}(a_0\cos x+a_1\sin x).$$ (16)

The Bessel Functions of order $\pm {1/2}$ are therefore defined as

$$J_{-1/2}(x) \equiv \sqrt{{2\over\pi x}} \cos x$$ (17)

$$J_{1/2}(x) \equiv \sqrt{{2\over\pi x}} \sin x,$$ (18)

so the general solution for $m=\pm {1/2}$ is

$$y = a_0'J_{-1/2}(x)+a_1'J_{1/2}(x).$$ (19)

Now, consider a general $m \not = - {1/2}$. Equation (9) requires

$$a_1(2m+1) =0$$ (20)

$$[a_nn(2m+n)+a_{n-2}]x^{m+n} = 0$$ (21)

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

$$a_1 = 0$$ (22)

$$a_n = - {1\over n(2m+n)} a_{n-2}$$ (23)

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

$$a_{2l+1} = -{1\over(2l+1)[2(m+1)+1]} a_{2l-1}$$

$$= \ldots = f(n,m) a_1 = 0,$$ (24)

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

$$a_{2l} = - {1\over 2l(2m+2l)} a_{2l-2} = - {1\over 4l(m+l)} a_{2l-2}$$

$$= {(-1)^l\over [4l(m+l)][4(l-1)(m+l-1)]\cdots [4\cdot (m+1)]} a_0.$$ (25)

Plugging back into (9),

$$y = \sum_{n=0}^\infty a_nx^{n+m} = \sum_{n=1,3,5,\ldots}^\infty a_nx^{n+m}+\sum_{n=0,2,4,\ldots}^\infty a_nx^{n+m}$$

$$= \sum_{l=0}^\infty a_{2l+1}x^{2l+m+1} + \sum_{l=0}^\infty a_{2l}x^{2l+m}$$

$$= a_0 \sum_{l=0}^\infty {(-1)^l\over [4l(m+l)][4(l-1)(m+l-1)]\cdots [4\cdot (m+1)]} x^{2l+m}$$

$$= a_0 \sum_{l=0}^\infty {[(-1)^lm(m-1)\cdots 1]x^{2l+m}\over [4l(m+l)][4(l-1)(m+l-1)]\cdots [m(m-1)\cdots 1]}$$

$$= a_0 \sum_{l=0}^\infty {(-1)^lm!\over 4^ll!(m+l)!} = a_0 \sum_{l=0}^\infty {(-1)^lm!\over 2^{2l}l!(m+l)!}.$$ (26)

Now define

$$J_m(x) \equiv \sum_{l=0}^\infty {(-1)^l\over 2^{2l+m}l!(m+l)!} x^{2l+m},$$ (27)

where the factorials can be generalized to Gamma Functions for nonintegral $m$. The above equation then becomes

$$y = a_0 2^m m! J_m(x) = a_0' J_m(x).$$ (28)

Returning to equation (5) and examining the case $k = -m$,

$$a_1(1-2m) + \sum_{n=2}^\infty [a_nn(n-2m)+a_{n-2}]x^{n-m} = 0.$$ (29)

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 $-\vert m\vert$, so

$$a_1(1+2\vert m\vert) + \sum_{n=2}^\infty [a_nn(n+2\vert m\vert)+a_{n-2}]x^{\vert m\vert+n} = 0,$$ (30)

and we obtain the same solutions as before, but with $m$ replaced by $\vert m\vert$.

$$J_m(x)=\cases{\sum_{l=0}^\infty {(-1)^l\over 2^{2l+\vert m\v... ...rt{{2\over\pi x}} \sin x & for $m={\textstyle{1\over 2}}$.\cr}$$ (31)

We can relate $J_m$ and $J_{-m}$ (when $m$ is an Integer) by writing

$$J_{-m}(x)=\sum_{l=0}^\infty {(-1)^l\over 2^{2l-m}l!(l-m)!} x^{2l-m}.$$ (32)

Now let $l\equiv l'+m$. Then

$$J_{-m}(x) = \sum_{l'+m=0}^\infty {(-1)^{l'+m}\over 2^{2l'+m}(l'+m)!l!} x^{2l'+m}$$

$$= \sum_{l'=-m}^{-1} {(-1)^{l'+m}\over 2^{2l'+m}l'!(l'+m)!} x^{2l'+m} + \sum_{l'=0}^\infty {(-1)^{l'+m}\over 2^{2l'+m}l'!(l'+m)!} x^{2l'+m}.$$ (33)

But $l'! =\infty$ for $l'=-m,\ldots,-1$, so the Denominator is infinite and the terms on the right are zero. We therefore have

$$J_{-m}(x)=\sum_{l=0}^\infty {(-1)^{l+m}\over 2^{2l+m}l!(l+m)!} x^{2l+m}=(-1)^m J_m(x).$$ (34)

Note that the Bessel Differential Equation is second-order, so there must be two linearly independent solutions. We have found both only for $\vert m\vert={1/2}$. 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

$$Z_m \equiv C_1J_m(x)+C_2Y_m(x),$$ (35)

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 $[0,1]$ with respect to the weight factor $x$. Except when $2n$ is a Negative Integer,

$$J_m(z) = {z^{-1/2}\over 2^{2m+1/2}i^{m+1/2}\Gamma(m+1)} M_{0,m}(2iz),$$ (36)

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

$$J_\nu(z) = {({1\over 2}z)^\nu\over \Gamma(\nu+1)} {}_0F_1(\nu+1; -{\textstyle{1\over 4}}z^2).$$ (37)

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

$$J_n(x)=i^n T_n\left({i{d\over dx}}\right)J_0(x),$$ (38)

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

$$J_m(x) \approx {1\over\Gamma(m+1)} \left({x\over 2}\right)^m$$ (39)

for $x\ll 1$ and

$$J_m(x) \approx \sqrt{{2\over\pi x}} \cos\left({x-{m\pi\over 2}-{\pi\over 4}}\right)$$ (40)

for $x\gg 1$. A derivative identity is

$${d\over dx} [x^mJ_m(x)] = x^mJ_{m-1}(x).$$ (41)

An integral identity is

$$\int^u_0 u'J_0(u')\, du' = uJ_1(u).$$ (42)

Some sum identities are

$$1 = [J_0(x)]^2+2[J_1(x)]^2+2[J_2(x)]^2+\ldots$$ (43)

$$1 = J_0(x)+2J_2(x)+2J_4(x)+\dots$$ (44)

and the Jacobi-Anger Expansion

$$e^{iz\cos\theta}=\sum_{n=-\infty}^\infty i^n J_n(z)e^{in\theta},$$ (45)

which can also be written

$$e^{iz\cos\theta}=J_0(z)+2\sum_{n=1}^\infty i^n J_n(z)\cos(n\theta).$$ (46)

The Bessel function addition theorem states

$$J_n(y+z)=\sum_{m=-\infty}^\infty J_m(y)J_{n-m}(z).$$ (47)

Roots of the Function $J_n(x)$ are given in the following table.

zero	$J_0(x)$	$J_1(x)$	$J_2(x)$	$J_3(x)$	$J_4(x)$	$J_5(x)$
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 $x_n$ be the $n$th Root of the Bessel function $J_0(x)$, then

$$\sum_{n=1}^\infty {1\over x_n J_0(x_n)}=0.38479\ldots$$ (48)

(Le Lionnais 1983).

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

zero	${J_0}'(x)$	${J_1}'(x)$	${J_2}'(x)$	${J_3}'(x)$	${J_4}'(x)$	${J_5}'(x)$
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

$$J_0(z) = {1\over 2\pi} \int^{2\pi}_0 e^{iz} \cos \phi\,d\phi$$ (49)

$$J_n(z) = {1\over\pi}\int_0^\pi \cos(z\sin\theta-n\theta)\,d\theta,$$ (50)

which is Bessel's First Integral,

$$J_n(z) = {i^{-n}\over\pi} \int_0^\pi e^{iz\cos\theta}\cos(n\theta)\, d\theta$$ (51)

$$J_n(z) = {1\over 2\pi i^n} \int^{2\pi}_0 e^{iz\cos \phi} e^{in\phi} \, d\phi$$ (52)

for $n=1$, 2, ...,

$$J_n(z) = {2\over\pi } {x^n\over (2m-1)!!} \int^{\pi/2}_0 \sin^{2n} u\cos(x \cos u)\,du$$ (53)

for $n=1$, 2, ...,

$$J_n(x) = {1\over 2\pi i} \int_\gamma e^{(x/2)(z-1/z)}z^{-n-1}\,dz$$ (54)

for $n > -1/2$. Integrals involving $J_1(x)$ include

$$\int_0^\infty J_1(x)\,dx = 1$$ (55)

$$\int_0^\infty \left[{J_1(x)\over x}\right]^2\,dx = {4\over 3\pi}$$ (56)

$$\int_0^\infty \left[{J_1(x)\over x}\right]^2 x\,dx = {1\over 2}.$$ (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 $J$ 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.

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 $J_0(x)$ 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.