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首页> 外文期刊>Journal of Computational Physics >On the nonoscillatory phase function for Legendre's differential equation
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On the nonoscillatory phase function for Legendre's differential equation

机译：关于Legendre的微分方程的非振动阶段函数

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AbstractWe express a certain complex-valued solution of Legendre's differential equation as the product of an oscillatory exponential function and an integral involving only nonoscillatory elementary functions. By calculating the logarithmic derivative of this solution, we show that Legendre's differential equation admits a nonoscillatory phase function. Moreover, we derive from our expression an asymptotic expansion useful for evaluating Legendre functions of the first and second kinds of large orders, as well as the derivative of the nonoscillatory phase function. Our asymptotic expansion is not as efficient as the well-known uniform asymptotic expansion of Olver; however, unlike Olver's expansion, it coefficients can be easily obtained. Numerical experiments demonstrating the properties of our asymptotic expansion are presented.]]>
机译：<！[cdata [ Abstract 我们表达了Legendre的差分方程的某个复值解决方案作为振荡指数函数的乘积和仅涉及非振动基本功能的积分。通过计算该解决方案的对数衍生物，我们表明Legendre的差动方程承认了非振动阶段函数。此外，我们从我们的表达中获得了一种用于评估第一和第二种大订单的传奇函数的渐近扩展，以及非阳离阶段函数的衍生物。我们的渐近扩张与奥尔弗的众所周知的均匀渐近扩张并不高效;但是，与Olver的扩展不同，可以轻松获得IT系数。介绍了表现出渐近扩张性质的数值实验。 ]]>

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