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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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