BESSEL-FUNCTIONS OF PURELY IMAGINARY ORDER, WITH AN APPLICATION TO 2ND-ORDER LINEAR-DIFFERENTIAL EQUATIONS HAVING A LARGE PARAMETER
BESSEL-FUNCTIONS OF PURELY IMAGINARY ORDER, WITH AN APPLICATION TO 2ND-ORDER LINEAR-DIFFERENTIAL EQUATIONS HAVING A LARGE PARAMETER
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DOI：
10.1137/0521055
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发表时间：
1990-07-01
影响因子：
2
通讯作者：
DUNSTER, TM
中科院分区：
文献类型：
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作者：
DUNSTER, TM
检查纯虚数阶的贝塞尔函数。定义了修正和未修正贝塞尔方程的解，当它们的阶数是纯虚数并且它们的参数是实数和正数时，它们是实数值令人满意的函数对。对于这些数值上令人满意的函数，推导了递推关系、解析连续公式、幂级数表示、朗斯基关系、积分表示、奇点行为和零点的渐近形式。此外，当贝塞尔函数的阶数为纯虚数且绝对值较大时，推导了初等函数和艾里函数的渐近展开式。然后，针对极点指数为复数的情况，检查具有大参数和简单极点的二阶线性常微分方程。根据纯虚数阶的数值令人满意的贝塞尔函数导出解的渐近展开式。
Bessel functions of purely imaginary order are examined. Solutions of both the modified and unmodified Bessel equations are defined which, when their order is purely imaginary and their argument is real and positive, are pairs of real numerically satisfactory functions. Recurrence relations, analytic continuation formulas, power series representations, Wronskian relations, integral representations, behavior at singularities, and asymptotic forms of the zeros are derived for these numerically satisfactory functions. Also, asymptotic expansions in terms of elementary and Airy functions are derived for the Bessel functions when their order is purely imaginary and of large absolute value.Second-order linear ordinary differential equations having a large parameter and a simple pole are then examined, for the case where the exponent of the pole is complex. Asymptotic expansions are derived for the solutions in terms of the numerically satisfactory Bessel functions of purely imaginary order.