Bessel's equation

The linear differential equation

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
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): $$ x^2\sum_{k=0}^\infty(r+k)(r+k-1)a_kx^{r+k-2}+ x\sum_{k=0}^\infty(r+k)a_kx^{r+k-1}+ (x^2-p^2)\sum_{k=0}^\infty a_kx^{r+k} \;=\; 0. $$ 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

The last of those can be written $$(r+k-p)(r+k+p)a_k+a_{k-2} \;=\; 0.$$ Because $a_0 \neq 0$ , the first of those (the indicial equation) gives $r^2-p^2 = 0$ , i.e. we have the roots $$r_1 \;=\; p, \quad r_2 \;=\; -p.$$ Let's first look the the solution of (1) with $r = p$ ; then $k(2p+k)a_k+a_{k-2} = 0$ , and thus $$a_k \;=\; -\frac{a_{k-2}}{k(2p+k).}$$ 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
(where $m = 1,\,2,\,\ldots$ ). Putting the obtained coefficients to (2) we get the particular solution

In order to get the coefficients $a_k$ for the second root $r_2 = -p$ we have to look after that $$(r_2+k)^2-p^2 \;\neq\; 0,$$ or $r_2+k \neq p = r_1$ . Therefore $$r_1-r_2 \;=\; 2p \;\neq\; k$$ 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 $-p$ :

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

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

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 $$y_1 \;=\; \frac{x^{\frac{1}{2}}}{\sqrt{2}\,\Gamma(\frac{3}{2})}\left[1\!-\!\frac{x^2}{2\!\cdot\!3}\!+\!\frac{x^4}{2\!\cdot\!4\!\cdot\!3\!\cdot\!5}\!-\!\frac{x^6}{2\!\cdot\!4\cdot\!6\!\cdot\!3\!\cdot\!5\!\cdot\!7}\!+-\ldots\right] \;=\; \sqrt{\frac{2}{\pi x}}\left(x\!-\!\frac{x^3}{3!}\!+\!\frac{x^5}{5!}\!-+\ldots\right).$$ Thus we get $$J_{\frac{1}{2}}(x) \;=\; \sqrt{\frac{2}{\pi x}}\sin{x};$$ analogically (6) yields $$J_{-\frac{1}{2}}(x) \;=\; \sqrt{\frac{2}{\pi x}}\cos{x},$$ and the general solution of the equation (1) for $p = \frac{1}{2}$ is $$y \;:=\; C_1J_{\frac{1}{2}}(x)+C_2J_{-\frac{1}{2}}(x).$$

In the case that $p$ is a non-negative integer $n$ , the ``+'' case of (7) gives the solution $$J_{n}(x) \;=\; \sum_{m=0}^\infty \frac{(-1)^m}{m!\,(m+n)!}\left(\frac{x}{2}\right)^{2m+n}, $$ but for $p = -n$ the expression of $J_{-n}(x)$ is $(-1)^nJ_n(x)$ , i.e. linearly dependent on $J_n(x)$ . 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 $F$ :

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 $$c_n \;=\; \frac{1}{2\pi i}\oint_{\gamma} \frac{f(t)}{(t-a)^{n+1}}\,dt$$ 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 $$c_n \;=\; \frac{1}{2\pi i} \oint_\gamma\frac{e^{\frac{zt}{2}}e^{-\frac{z}{2t}}}{t^{n+1}}\,dt \;=\; \frac{1}{2\pi i}\left(\frac{z}{2}\right)^n\! \oint_\delta\frac{e^\zeta e^{-\frac{z^2}{4\zeta}}}{\zeta^{n+1}}\,d\zeta \;=\; \sum_{m=0}^\infty\frac{(-1)^m}{m!}\left(\frac{z}{2}\right)^{2m+n}\! \frac{1}{2\pi i}\oint_\delta \zeta^{-m-n-1}e^\zeta\,d\zeta.$$ 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 $$c_n \;=\; \sum_{m=0}^\infty \frac{(-1)^m}{m!\Gamma(m+n+1)}\left(\frac{z}{2}\right)^{2m+n} \;=\; J_n(z).$$ 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: $$J_n(z) \;=\; \frac{1}{\pi}\int_0^\pi\cos(n\varphi-z\sin{\varphi})\,d\varphi$$ Also one can obtain the addition formula $$J_n(x\!+\!y) \;=\; \sum_{\nu=-\infty}^{\infty}J_\nu(x)J_{n-\nu}(y)$$ and the series representations of cosine and sine: $$\cos{z} \;=\; J_0(z)-2J_2(z)+2J_4(z)-+\ldots$$ $$\sin{z} \;=\; 2J_1(z)-2J_3(z)+2J_5(z)-+\ldots$$

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).