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[parent] Bessel equation (Example)

The linear differential equation

$\displaystyle x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+(x^2-p^2)y = 0,$ (1)
in which $p$ is a constant (non-negative if it is real), is called the Bessel's equation.  We derive its general solution by trying the series form
$\displaystyle y = x^r\sum_{k=0}^\infty a_kx^k = \sum_{k=0}^\infty a_kx^{r+k},$ (2)
due to Frobenius.  Since the parameter $r$ is indefinite, we may regard $a_0$ as distinct from 0.

We substitute (2) and the derivatives of the series in (1):

$\displaystyle x^2\sum_{k=0}^\infty(r+k)(r+k-1)a_kx^{r+k-2}+ x\sum_{k=0}^\infty(r+k)a_kx^{r+k-1}+ (x^2-p^2)\sum_{k=0}^\infty a_kx^{r+k} = 0. $
Thus the coefficients of the powers $x^r$, $x^{r+1}$, $x^{r+2}$ and so on must vanish, and we get the system of equations
\begin{align*}\begin{cases} {[}r^2-p^2{]}a_0 = 0,\ {[}(r+1)^2-p^2{]}a_1 = 0,\\... ...qquad \qquad \ldots\ {[}(r+k)^2-p^2{]}a_k+a_{k-2} = 0. \end{cases}\end{align*} (3)
The last of those can be written

$\displaystyle (r+k-p)(r+k+p)a_k+a_{k-2} = 0.$
Because  $a_0 \neq 0$,  the first of those (the indicial equation) gives  $r^2-p^2 = 0$,  i.e. we have the roots

$\displaystyle r_1 = p,\,\, r_2 = -p.$
Let's first look the the solution of (1) with  $r = p$;  then  $k(2p+k)a_k+a_{k-2} = 0$,  and thus 

$\displaystyle a_k = -\frac{a_{k-2}}{k(2p+k).}$
From the system (3) we can solve one by one each of the coefficients $a_1$, $a_2$, $\ldots$  and express them with $a_0$ which remains arbitrary.  Setting for $k$ the integer values we get
\begin{align*}\begin{cases} a_1 = 0,\,\,a_3 = 0,\,\ldots,\, a_{2m-1} = 0;\ a_2... ...^ma_0}{2\cdot4\cdot6\cdots(2m)(2p+2)(2p+4)\ldots(2p+2m)} \end{cases}\end{align*} (4)
(where  $m = 1,\,2,\,\ldots$). Putting the obtained coefficients to (2) we get the particular solution
$\displaystyle y_1 := a_0x^p \left[1\!\!\frac{x^2}{2(2p\!+\!2)}\! +\!\frac{x^4}{... ...frac{x^6}{2\!\cdot\!4\!\cdot\!6(2p\!+\!2)(2p\!+\!4)(2p\!+\!6)}\!+-\ldots\right]$ (5)

In order to get the coefficients $a_k$ for the second root  $r_2 = -p$  we have to look after that

$\displaystyle (r_2+k)^2-p^2 \neq 0,$
or  $r_2+k \neq p = r_1$.  Therefore

$\displaystyle r_1-r_2 = 2p \neq k$
where $k$ is a positive integer.  Thus, when $p$ is not an integer and not an integer added by $\frac{1}{2}$, we get the second particular solution, gotten of (5) by replacing $p$ by $-p$:
$\displaystyle y_2 := a_0x^{-p}\!\left[1 \!-\!\frac{x^2}{2(-2p\!+\!2)}\!+\!\frac... ...c{x^6}{2\!\cdot\!4\!\cdot\!6(-2p\!+\!2)(-2p\!+\!4)(-2p\!+\!6)}\!+-\ldots\right]$ (6)

The power series of (5) and (6) converge for all values of $x$ and are linearly independent (the ratio $y_1/y_2$ tends to 0 as  $x\to\infty$).  With the appointed value

$\displaystyle a_0 = \frac{1}{2^p\,\Gamma(p+1)},$
the solution $y_1$ is called the Bessel function of the first kind and of order $p$ and denoted by $J_p$.  The similar definition is set for the first kind Bessel function of an arbitrary order  $p\in \mathbb{R}$ (and $\mathbb{C}$). For  $p\notin \mathbb{Z}$  the general solution of the Bessel's differential equation is thus

$\displaystyle y := C_1J_p(x)+C_2J_{-p}(x),$
where  $J_{-p}(x) = y_2$  with  $a_0 = \frac{1}{2^{-p}\Gamma(-p+1)}$.

The explicit expressions for $J_{\pm p}$ are

$\displaystyle J_{\pm p}(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m!\,\Gamma(m\pm p+1)}\left(\frac{x}{2}\right)^{2m\pm p},$ (7)
which are obtained from (5) and (6) by using the last formula for gamma function.

E.g. when  $p = \frac{1}{2}$  the series in (5) gets the form

$\displaystyle y_1 = \frac{x^{\frac{1}{2}}}{\sqrt{2}\,\Gamma(\frac{3}{2})}\left[... ...\frac{2}{\pi x}}\left(x\!-\!\frac{x^3}{3!}\!+\!\frac{x^5}{5!}\!-+\ldots\right).$
Thus we get

$\displaystyle J_{\frac{1}{2}}(x) = \sqrt{\frac{2}{\pi x}}\sin{x};$
analogically (6) yields

$\displaystyle J_{-\frac{1}{2}}(x) = \sqrt{\frac{2}{\pi x}}\cos{x},$
and the general solution of the equation (1) for  $p = \frac{1}{2}$  is

$\displaystyle y := C_1J_{\frac{1}{2}}(x)+C_2J_{-\frac{1}{2}}(x).$

In the case that $p$ is a non-negative integer $n$, the “+” case of (7) gives the solution

$\displaystyle J_{n}(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m!\,(m+n)!}\left(\frac{x}{2}\right)^{2m+n}, $
but for  $p = -n$  the expression of $J_{-n}(x)$ is $(-1)^nJ_n(x)$, i.e. linearly dependent of $J_n(x)$.  It can be shown that the other solution of (1) ought to be searched in the form  $y = K_n(x) = J_n(x)\ln{x}+x^{-n}\sum_{k=0}^\infty b_kx^k$.  Then the general solution is  $y := C_1J_n(x)+C_2K_n(x)$.

Other formulae

The first kind Bessel functions of integer order have the generating function $F$:

$\displaystyle F(z,\,t) = e^{\frac{z}{2}(t-\frac{1}{t})} = \sum_{n=-\infty}^\infty J_n(z)t^n$ (8)
This function has an essential singularity at  $t = 0$  but is analytic elsewhere in $\mathbb{C}$; thus $F$ has the Laurent expansion in that point.  Let us prove (8) by using the general expression

$\displaystyle c_n = \frac{1}{2\pi i}\oint_{\gamma} \frac{f(t)}{(t-a)^{n+1}}\,dt$
of the coefficients of Laurent series.  Setting to this  $a := 0$,  $f(t) := e^{\frac{z}{2}(t-\frac{1}{t})}$,  $\zeta := \frac{zt}{2}$  gives

$\displaystyle c_n = \frac{1}{2\pi i} \oint_\gamma\frac{e^{\frac{zt}{2}}e^{-\fra... ...{2}\right)^{2m+n}\! \frac{1}{2\pi i}\oint_\delta \zeta^{-m-n-1}e^\zeta\,d\zeta.$
The paths $\gamma$ and $\delta$ go once round the origin anticlockwise in the $t$-plane and $\zeta$-plane, respectively.  Since the residue of $\zeta^{-m-n-1}e^\zeta$ in the origin is  $\frac{1}{(m+n)!} = \frac{1}{\Gamma(m+n+1)}$,  the residue theorem gives

$\displaystyle c_n = \sum_{m=0}^\infty \frac{(-1)^m}{m!\Gamma(m+n+1)}\left(\frac{z}{2}\right)^{2m+n} = J_n(z).$
This means that $F$ has the Laurent expansion (8).

By using the generating function, one can easily derive other formulae, e.g. the integral representation of the Bessel functions of integer order:

$\displaystyle J_n(z) = \frac{1}{\pi}\int_0^\pi\cos(n\varphi-z\sin{\varphi})\,d\varphi$
Also one can obtain the addition formula

$\displaystyle J_n(x+y) = \sum_{\nu=-\infty}^{\infty}J_\nu(x)J_{n-\nu}(y)$
and the series representations of cosine and sine:

$\displaystyle \cos{z} = J_0(z)-2J_2(z)+2J_4(z)-+\ldots$

$\displaystyle \sin{z} = 2J_1(z)-2J_3(z)+2J_5(z)-+\ldots$

Bibliography

1
N. PISKUNOV: Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele.  Kirjastus Valgus, Tallinn (1966).
2
K. KURKI-SUONIO: Matemaattiset apuneuvot.  Limes r.y., Helsinki (1966).



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This is version 1 of Bessel equation, born on 2009-04-19.
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Physics Classification: 02.30.Hq (Ordinary differential equations)
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