从振动波形研究贝塞尔函数和勒让德函数的渐近关系

Study of asymptotic relationship between Bessel function and Legendre function by analyzing vibration modes

通过圆平面和球面上对称振动的分析,比较了零阶贝塞尔函数和勒让德函数曲线,发现两者形状有明显的相似性.在进行振幅变换后,两者曲线几乎重合.从数学和物理角度解释了圆平面和球面上振动波形相似的原因,导出了零阶贝塞尔函数和勒让德函数的渐近关系,并得到了贝塞尔函数零点的一个估计公式.

By analyzing the symmetric vibration on a disk and a sphere surface, the curves of the zero-order Bessel function and the Legendre function are compared. It is shown that there are apparent similarities between them. After transformation of amplitude, the curves of the Bessel function agree with those of the Legendre function very well. The reasons for the similarities are explained both in mathematical and physical approaches. The asymptotic relationship between the functions of two kinds and an estimating formula for zeros of the Bessel function are derived.