Uniform asymptotic expansions of integrals with stationary point near algebraic singularity
Uniform asymptotic expansions of integrals with stationary point near algebraic singularity
复制标题
代数奇点附近驻点积分的一致渐近展开
DOI：
10.1002/cpa.3160190403
复制
发表时间：
1966
影响因子：
3
通讯作者：
N. Bleistein
中科院分区：
文献类型：
--
作者：
N. Bleistein
本文考虑由以下类型的积分定义的函数Z （A； A）。假设f在t= 01处有一个简单的驻点，即& (A； A)= 0, ftt (A； A)# 0，我们希望确定I对大值il的渐近展开式。通常可以用下列方法之一求得膨胀：1)拉普拉斯法，2)固定相法，3)最陡下降法（见[1]）。但是在我们的例子中，被积函数在t= 0处有一个奇点或者零点。这些方法对不同的2值产生不同的结果，这取决于驻点是远离奇点还是正好在奇点上。我们希望得到对任意位置的驻点相对于奇点ix有效的展开式。，膨胀在x…在衍射问题中出现Z （l; a）型积分。大参数为kr， x= 0对应某个临界反射角或散射角；参见[Z]。
In this paper we shall consider functions Z (A; a) defined by integrals of the typeWe wish to determine the asymptotic expansion of I for large values of il assuming that f has a simple stationary point at t= 01, ie, & (a; a)= 0, ftt (a; a)# 0. Normally the expansion might be obtained by one of the following methods: 1) Laplace’s method, 2) the method of stationary phase, 3) the method of steepest descent (see [I]). However in our case the integrand has a singularity or a zero at t= 0. These methods yield different kinds of results for different values of 2, depending on whether the stationary point is away from the singular point or right at it. We wish to obtain an expansion valid for any location of the stationary point relative to the singular point, ix., an expansion which is uniform in x.. l Integrals of the type of Z (L; a) arise in diffraction problems. The large parameter is kr and x= 0 corresponds to some critical angle of reflection or scattering; see for example [Z].