Bessel's Equation and Bessel Functions

Definition Let $\nu \ge 0$ be a real constant. Bessel’s equation of order $\nu$ is

where $x$ is the independent variable and $y' = dy/dx$. The constant $\nu$ is called the order, and the solutions are the Bessel functions of order $\nu$. This order is a parameter of the equation and not its differential order, which is $2$ for every $\nu$. Only $\nu^2$ appears, so the orders $\nu$ and $-\nu$ give the same equation, and the convention $\nu \ge 0$ loses nothing. Dividing by $x^2$ gives the normalized coefficients $p(x) = 1/x$ of $y'$ and $q(x) = 1 - \nu^2/x^2$ of $y$. Every $x \ne 0$ is an ordinary point, so the regular intervals are $(-\infty, 0)$ and $(0, \infty)$, and solutions are usually studied for $x > 0$. The point $x = 0$ is a regular singular point, because $x\,p(x) = 1$ and $x^2q(x) = x^2 - \nu^2$ are analytic there. Its indicial equation $r(r - 1) + r - \nu^2 = 0$ simplifies to $r^2 = \nu^2$, with the roots $r = \nu$ and $r = -\nu$, so the Frobenius method applies at the origin.

Definition For $\nu \ge 0$, the Frobenius solution with the exponent $r = \nu$, scaled in the standard way, is the Bessel function of the first kind of order $\nu$,

where $\Gamma$ is the gamma function, which satisfies $\Gamma(m + 1) = m!$ for every integer $m \ge 0$. The ratio of consecutive terms is $-x^2/(4(k + 1)(k + \nu + 1))$, which tends to $0$, so the series converges for every $x > 0$ and solves Bessel’s equation of order $\nu$ there. For an integer order it is a power series that converges and solves the equation on the whole line. For a non-integer $\nu > 0$, the same formula with $\nu$ replaced by $-\nu$ defines $J_{-\nu}$. It is the Frobenius solution with the exponent $-\nu$ and also solves the equation for $x > 0$.

Definition For a non-integer $\nu$, the functions $J_\nu$ and $J_{-\nu}$ are linearly independent on $(0, \infty)$, since their Wronskian is

For such $\nu$ the Bessel function of the second kind of order $\nu$ is the combination

and for an integer $n \ge 0$ it is the limit $Y_n(x) = \lim_{\nu \to n} Y_\nu(x)$. For every $\nu \ge 0$ the function $Y_\nu$ solves Bessel’s equation of order $\nu$ on $(0, \infty)$, and

So $J_\nu$ and $Y_\nu$ form a fundamental set on $(0, \infty)$, and every solution there is $y = C_1J_\nu(x) + C_2Y_\nu(x)$. Abel’s identity predicts the shape of this Wronskian, since $p(x) = 1/x$ gives $W(x) = C/x$, and the standard scaling of $Y_\nu$ makes $C = 2/\pi$. The limit is needed at integer orders because $J_{-n}$ brings nothing new there. Read with the convention $1/\Gamma(m) = 0$ for $m = 0, -1, -2, \dots$, the series for $J_{-n}$ loses its first $n$ terms and reduces to $J_{-n} = (-1)^nJ_n$, and the Wronskian above is $0$ at every integer $\nu$.

Definition Write $f \sim g$ when $f(x)/g(x) \to 1$. As $x \to 0^+$,

the first for every $\nu \ge 0$ and the last for $\nu > 0$. So $J_0(x) \to 1$, $J_\nu(x) \to 0$ for $\nu > 0$, and every $Y_\nu$ is unbounded near $0$. Consequently, on an interval $(0, b)$, a solution of Bessel’s equation stays bounded as $x \to 0^+$ exactly when it is a constant multiple of $J_\nu$. Indeed, if $y = C_1J_\nu + C_2Y_\nu$ with $C_2 \ne 0$, then $y/Y_\nu = C_1J_\nu/Y_\nu + C_2$ tends to $C_2 \ne 0$, so $\lvert y \rvert$ grows without bound.

Definition At order $\nu = \tfrac{1}{2}$ the Bessel functions are elementary. For $x > 0$,

and $Y_{1/2} = -J_{-1/2}$, from the definition of the second kind with $\cot(\pi/2) = 0$ and $\csc(\pi/2) = 1$. The reason is the substitution $y = u/\sqrt{x}$, which turns Bessel’s equation of order $\nu$ into

For $\nu = \tfrac{1}{2}$ the fraction vanishes, which leaves $u'' + u = 0$ with the solutions $\sin x$ and $\cos x$. Every order $\nu = m + \tfrac{1}{2}$ with an integer $m \ge 0$ also gives elementary functions built from $\sin x$, $\cos x$ and powers of $x$, for example

These orders appear when the wave or heat equation is separated in spherical coordinates.

Definition In polar coordinates $(r, \theta)$ the Laplacian of a function $u(r, \theta)$ is

Let $\kappa > 0$ be a constant. A separated solution $u = R(r)\Theta(\theta)$ of the Helmholtz equation $\Delta u + \kappa^2u = 0$, with primes meaning $d/d\theta$ on $\Theta$ and $d/dr$ on $R$, leads to $\Theta'' + \mu\Theta = 0$ for a separation constant $\mu$. A nonzero $\Theta$ with period $2\pi$ exists only for $\mu = m^2$ with an integer $m \ge 0$, and the radial factor then satisfies

Writing $R(r) = y(\kappa r)$ with $x = \kappa r$ turns this into Bessel’s equation of order $m$ for $y$, so $R(r) = C_1J_m(\kappa r) + C_2Y_m(\kappa r)$ for $r > 0$. On a full disk $0 \le r \le a$, a solution that stays bounded at the center needs $C_2 = 0$. On an annulus $b \le r \le a$ with $b > 0$ the center is not in the region, $Y_m(\kappa r)$ is bounded there, and both kinds are allowed, with the boundary conditions at $r = b$ and $r = a$ deciding the combination. The Helmholtz equation governs the shape of a vibration mode. A mode $u = U(r, \theta)\cos(\omega t)$ of the wave equation $u_{tt} = c^2\Delta u$ needs $\Delta U + \kappa^2U = 0$ with $\kappa = \omega/c$.