Chapter 18

Bessel equations.

As introduced in the textbook, the Bessel equation is given by $$t^2 y'' + t y' + (t^2 - n^2) y = 0$$ (with $n = 0, 1, 2, \dots$).

Let's first focus on the Bessel equation with $n=0$: \begin{equation}\label{bessel0} t^2 y'' + t y' + t^2 y = 0,\,\,\,\mbox{ or equivalently } \,\,\, y'' + \frac{1}{t}y' + y = 0. \end{equation}

Using power series, we find a solution of the following form, which is called the Bessel function $J_0(t)$: $$J_0(t) = \sum_{k=0}^{\infty} \frac{(-1)^k}{2^{2k} (k!)^2} t^{2k} = 1 - \frac{1}{4}t^2 + \frac{1}{64}t^4 - \frac{1}{2304} t^6 + \dots .$$

It can be shown that the scalar multiples $c \, J_0(t)$ are the only solutions of $t^2 y'' + t y' + t^2 y = 0$ on $(0, +\infty)$ that remain bounded as $t \to 0^+$. We have $c\,J_0(t) \to c$ as $t \to 0^+$.

It is tempting to think that since $J_0(t)$ satisfies the ODE $y'' + (1/t)y' + y = 0$ (as on the right in the Bessel equation and for large $t$ perhaps the middle term $(1/t)y'$ is "negligible", the behavior of $J_0(t)$ for large $t$ should be approximated by some nonzero solution to $y'' + y = 0$; i.e., a nonzero linear combination of $\sin t$ and $\cos t$. But that guess is wrong! Such linear combinations $c_1 \cos t + c_2 \sin t$ are phase-shifted cosine waves $A \cos(t - \phi)$ up to a change in amplitude, and in particular they do not decay as $t \to +\infty$, whereas it can be shown that $J_0(t)$ decays like $1/\sqrt{t}$, and more precisely \begin{equation}\label{J0bigt} J_0(t) \approx \sqrt{\frac{2}{\pi t}} \cos(t - \pi/4) \end{equation} as $t \to +\infty$ (with error bounded by a constant multiple of $1/t^{3/2}$).

There is an analogous solution to $t^2 y'' + ty' + (t^2 - n^2) y = 0$ for positive integers $n$, denoted $J_n(t)$ and called the Bessel function of the "first kind" of order $n$. It is defined by a convergent power series $$J_n(t) = t^n \sum_{k=0}^{\infty} \frac{(-1)^k}{2^{2k+n}k!(k+n)!}t^{2k}$$ and satisfies an analogue of decaying for big $t$, with precise form depending on whether $n$ is even or odd: $$J_{2m}(t) \approx (-1)^m \sqrt{\frac{2}{\pi t}} \cos(t - \pi/4),$$ $$J_{2m+1}(t) \approx (-1)^m \sqrt{\frac{2}{\pi t}} \sin(t - \pi/4),$$ as $t \to +\infty$ (with error bounded by a constant multiple of $1/t^{3/2}$).

Let $J_{n,2r+n}(t) := t^n\sum_{k=0}^{r} \sum_{k=0}^{\infty} \frac{(-1)^k}{2^{2k+n}k!(k+n)!}t^{2k}$ be the sum up to the $(2r+n)$-th power term in the expansion of $J_n$.

Below we visualize the appriximations of the Bessel function $J_n$ both by $J_{n,2r+n}(t)$ and by the big $t$ asymptotic behavior ($(-1)^m\sqrt{\frac{2}{\pi t}} \cos(t - \pi/4)$ for even $n = 2m$ and $(-1)^m \sqrt{\frac{2}{\pi t}} \sin(t - \pi/4)$ for odd $n= 2m+1$). The red curve is this trigonometric function that approximates $J_n(t)$ for large $t$. You can change the visibility of a curve by clicking on its legend.