贝塞尔函数
以下微分方程的解所对应的函数 —
\begin{align} \label{eq:11101a} x^2 \, \frac{d^2y}{dx^2} + x \, \frac{dy}{dx} + (x^2 - p^2) \, y = 0 \end{align}
其中 —
| p | = 一个复数（贝塞尔函数的阶数） |
下图给出了两个常见的贝塞尔函数。注意，\( J_0(x) \) 类似于衰减的余弦，而 \( J_1(x) \) 类似于衰减的正弦。
|
|
|
修正贝塞尔函数
以下微分方程的解所对应的函数 —
\begin{align} \label{eq:11102a} x^2 \, \frac{d^2y}{dx^2} + x \, \frac{dy}{dx} - (x^2 + p^2) \, y = 0 \end{align}
下图给出了两个常见的修正贝塞尔函数。注意，\( I_0(x) \) 类似于双曲余弦，而 \( I_1(x) \) 类似于双曲正弦。
|
|
|
贝塞尔函数和修正贝塞尔函数可描述圆盘的弯曲振动与径向振动，以及无限长圆柱体内压力波的轴向振幅（以及许多其他物理现象）。
参考文献：
Abramowitz，第 355 - 434 页。
Kinsler，第 449 - 452 页 — 包含贝塞尔函数 \( J_0(x) \)、\( J_1(x) \) 和 \( J_2(x) \) 以及修正贝塞尔函数 \( I_0(x) \)、\( I_1(x) \) 和 \( I_2(x) \) 的函数值表
Wikipedia
Bessel function
A function that is the solution to the differential equation —
\begin{align} \label{eq:11101a} x^2 \, \frac{d^2y}{dx^2} + x \, \frac{dy}{dx} + (x^2 - p^2) \, y = 0 \end{align}
where —
| p | = a complex number (the order of the Bessel function) |
Two common Bessel functions are shown in the following graph. Note that \( J_0(x) \) resembles a decaying cosine while \( J_1(x) \) resembles a decaying sine.
|
|
|
Modified Bessel function
A function that is the solution to the differential equation —
\begin{align} \label{eq:11102a} x^2 \, \frac{d^2y}{dx^2} + x \, \frac{dy}{dx} - (x^2 + p^2) \, y = 0 \end{align}
Two common modified Bessel functions are shown in the following graph. Note that \( I_0(x) \) resembles a hyperbolic cosine while \( I_1(x) \) resembles a hyperbolic sine.
|
|
|
Bessel functions and modified Bessel functions describe the flexural and radial vibration of disks and the axial amplitude of a pressure wave in an infinitely long cylinder (and many other physical phenomena).
References:
Abramowitz, pp. 355 - 434.
Kinsler, pp. 449 - 452 — includes tables for Bessel functions \( J_0(x) \), \( J_1(x) \), and \( J_2(x) \) and modified Bessel functions \( I_0(x) \), \( I_1(x) \), and \( I_2(x) \)
Wikipedia