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Bessel Function of the First Kind

\begin{figure}\begin{center}\BoxedEPSF{BesselJ.epsf scaled 900}\end{center}\end{figure}

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
\end{displaymath} (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}.
\end{displaymath} (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.
\end{displaymath} (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
\end{displaymath} (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
\end{displaymath} (7)

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

a_1(2m+1) + \sum_{n=2}^\infty [a_nn(2m+n)+a_{n-2}]x^{m+n} = 0.
\end{displaymath} (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,
\end{displaymath} (10)

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

Now let $n \equiv 2l$, where $l = 1$, 2, ....
$\displaystyle a_{2l}$ $\textstyle =$ $\displaystyle - {1\over 2l(2l-1)} a_{2l-2}$  
  $\textstyle =$ $\displaystyle {(-1)^l\over[2l(2l-1)][2(l-1)(2l-3)]\cdots[2\cdot 1\cdot 1]} a_0$  
  $\textstyle =$ $\displaystyle {(-1)^l\over 2^ll!(2l-1)!!} a_0,$ (12)

which, using the identity $2^ll!(2l-1)!! = (2l)!$, gives
a_{2l} = {(-1)^l\over (2l)!} a_0.
\end{displaymath} (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,
\end{displaymath} (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.
\end{displaymath} (15)

Plugging back into (2) with $k = m = -{1/2}$ gives
$\displaystyle y$ $\textstyle =$ $\displaystyle x^{-1/2}\sum_{n=0}^\infty a_nx^n$  
  $\textstyle =$ $\displaystyle x^{-1/2}\left[{\,\sum_{n=1,3,5,\ldots}^\infty a_nx^n+\sum_{n=0,2,4,\ldots}^\infty a_nx^n}\right]$  
  $\textstyle =$ $\displaystyle x^{-1/2}\left[{\,\sum_{l=0}^\infty a_{2l}x^{2l}+\sum_{l=0}^\infty a_{2l+1}x^{2l+1}}\right]$  
  $\textstyle =$ $\displaystyle x^{-1/2}\left[{a_0 \sum_{l=0}^\infty {(-1)^l\over(2l)!} x^{2l}+a_1\sum_{l=0}^\infty {(-1)^l\over(2l+1)!}x^{2l+1}}\right]$  
  $\textstyle =$ $\displaystyle x^{-1/2}(a_0\cos x+a_1\sin x).$ (16)

The Bessel Functions of order $\pm {1/2}$ are therefore defined as
$\displaystyle J_{-1/2}(x)$ $\textstyle \equiv$ $\displaystyle \sqrt{{2\over\pi x}} \cos x$ (17)
$\displaystyle J_{1/2}(x)$ $\textstyle \equiv$ $\displaystyle \sqrt{{2\over\pi x}} \sin x,$ (18)

so the general solution for $m=\pm {1/2}$ is
y = a_0'J_{-1/2}(x)+a_1'J_{1/2}(x).
\end{displaymath} (19)

Now, consider a general $m \not = - {1/2}$. Equation (9) requires
a_1(2m+1) =0
\end{displaymath} (20)

\begin{displaymath}[a_nn(2m+n)+a_{n-2}]x^{m+n} = 0
\end{displaymath} (21)

for $n=2$, 3, ..., so
$\displaystyle a_1$ $\textstyle =$ $\displaystyle 0$ (22)
$\displaystyle a_n$ $\textstyle =$ $\displaystyle - {1\over n(2m+n)} a_{n-2}$ (23)

for $n=2$, 3, .... Let $n\equiv 2l+1$, where $l = 1$, 2, ..., then
$\displaystyle a_{2l+1}$ $\textstyle =$ $\displaystyle -{1\over(2l+1)[2(m+1)+1]} a_{2l-1}$  
  $\textstyle =$ $\displaystyle \ldots = f(n,m) a_1 = 0,$ (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

$\displaystyle a_{2l}$ $\textstyle =$ $\displaystyle - {1\over 2l(2m+2l)} a_{2l-2} = - {1\over 4l(m+l)} a_{2l-2}$  
  $\textstyle =$ $\displaystyle {(-1)^l\over [4l(m+l)][4(l-1)(m+l-1)]\cdots [4\cdot (m+1)]} a_0.$ (25)

Plugging back into (9),

$\displaystyle y$ $\textstyle =$ $\displaystyle \sum_{n=0}^\infty a_nx^{n+m} = \sum_{n=1,3,5,\ldots}^\infty a_nx^{n+m}+\sum_{n=0,2,4,\ldots}^\infty a_nx^{n+m}$  
  $\textstyle =$ $\displaystyle \sum_{l=0}^\infty a_{2l+1}x^{2l+m+1} + \sum_{l=0}^\infty a_{2l}x^{2l+m}$  
  $\textstyle =$ $\displaystyle a_0 \sum_{l=0}^\infty {(-1)^l\over [4l(m+l)][4(l-1)(m+l-1)]\cdots [4\cdot (m+1)]} x^{2l+m}$  
  $\textstyle =$ $\displaystyle a_0 \sum_{l=0}^\infty {[(-1)^lm(m-1)\cdots 1]x^{2l+m}\over [4l(m+l)][4(l-1)(m+l-1)]\cdots [m(m-1)\cdots 1]}$  
  $\textstyle =$ $\displaystyle a_0 \sum_{l=0}^\infty {(-1)^lm!\over 4^ll!(m+l)!} = a_0 \sum_{l=0}^\infty {(-1)^lm!\over 2^{2l}l!(m+l)!}.$ (26)

Now define
J_m(x) \equiv \sum_{l=0}^\infty {(-1)^l\over 2^{2l+m}l!(m+l)!} x^{2l+m},
\end{displaymath} (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).
\end{displaymath} (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.
\end{displaymath} (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,
\end{displaymath} (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}
\end{displaymath} (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}.
\end{displaymath} (32)

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

$\displaystyle J_{-m}(x)$ $\textstyle =$ $\displaystyle \sum_{l'+m=0}^\infty {(-1)^{l'+m}\over 2^{2l'+m}(l'+m)!l!} x^{2l'+m}$  
  $\textstyle =$ $\displaystyle \sum_{l'=-m}^{-1} {(-1)^{l'+m}\over 2^{2l'+m}l'!(l'+m)!} x^{2l'+m} + \sum_{l'=0}^\infty {(-1)^{l'+m}\over 2^{2l'+m}l'!(l'+m)!} x^{2l'+m}.$ (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).
\end{displaymath} (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),
\end{displaymath} (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),
\end{displaymath} (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).
\end{displaymath} (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),
\end{displaymath} (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
\end{displaymath} (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)
\end{displaymath} (40)

for $x\gg 1$. A derivative identity is
{d\over dx} [x^mJ_m(x)] = x^mJ_{m-1}(x).
\end{displaymath} (41)

An integral identity is
\int^u_0 u'J_0(u')\, du' = uJ_1(u).
\end{displaymath} (42)

Some sum identities are
1 = [J_0(x)]^2+2[J_1(x)]^2+2[J_2(x)]^2+\ldots
\end{displaymath} (43)

1 = J_0(x)+2J_2(x)+2J_4(x)+\dots
\end{displaymath} (44)

and the Jacobi-Anger Expansion
e^{iz\cos\theta}=\sum_{n=-\infty}^\infty i^n J_n(z)e^{in\theta},
\end{displaymath} (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).
\end{displaymath} (46)

The Bessel function addition theorem states
J_n(y+z)=\sum_{m=-\infty}^\infty J_m(y)J_{n-m}(z).
\end{displaymath} (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
\end{displaymath} (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

$\displaystyle J_0(z)$ $\textstyle =$ $\displaystyle {1\over 2\pi} \int^{2\pi}_0 e^{iz} \cos \phi\,d\phi$ (49)
$\displaystyle J_n(z)$ $\textstyle =$ $\displaystyle {1\over\pi}\int_0^\pi \cos(z\sin\theta-n\theta)\,d\theta,$ (50)

which is Bessel's First Integral,
$\displaystyle J_n(z)$ $\textstyle =$ $\displaystyle {i^{-n}\over\pi} \int_0^\pi e^{iz\cos\theta}\cos(n\theta)\, d\theta$ (51)
$\displaystyle J_n(z)$ $\textstyle =$ $\displaystyle {1\over 2\pi i^n} \int^{2\pi}_0 e^{iz\cos \phi} e^{in\phi} \, d\phi$ (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
\end{displaymath} (53)

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

for $n > -1/2$. Integrals involving $J_1(x)$ include
\int_0^\infty J_1(x)\,dx = 1
\end{displaymath} (55)

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

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


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.

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© 1996-9 Eric W. Weisstein