解析函数

By using Wigner distribution function, the beam combination characteristics of general astigmatic Gaussian beams are studied.

利用Wigner分布函数，对复杂像散高斯光束的并合问题进行了详细研究，得到了相干和非相干并合光束的光强分布和M2 因子的解析表达式。

By adopting the asymptotic expansion method, the higher order partial differential equation was transformed into the system of ordinary differential equations.

采用渐近展开法，将该控制方程——高阶偏微分方程转化为常微分方程组，求解该方程组获得应力函数，进而得到功能梯度材料中裂纹尖端应力高阶渐近场的解析式。

In the forth Chapter, we study the mean value of properties of Smarandache factorial function and it s mean value, using analytic methods give several interesting asymptotic formula for the reciprocal of S_k function.

在第四章中，本文研究了Smarandache高阶乘函数的性质及其均值估计问题，利用解析的方法得到一系列的渐进公式。

In Chapter 1, two solutions to the elliptical boundary value problem have been constructed through a monotone iterative process, and they might be identical. In Chapter 2, the author expresses the local solution to the parabolic initial boundary value problem taking advantage of Green function. In Chapter 3, the asymptotic solution to the initial value problem of Duffing equation is obtained and the method for finding the approximate solution to the inverse problem is put forward.This paper is planned with a view to the author's research means.

从研究结果看：第一章通过单调迭代过程构造出椭圆型方程的边值问题的两个解，并且指出二者可能是同一个解；第二章利用格林函数给出了抛物型方程的初边值问题的局部解的解析表达式；第三章中，先对Duffing方程的初值问题写出解的渐近展开式，再对其反问题提出一种求近似解的方法。

These achievements show the advantage of the ATF element again.

这些成果的取得再一次显示出解析试函数广义协调元的优势。

Numerical examples show that the ATF elements exhibit excellent performance and some traditional difficult problems appeared in the development of the FEM are solved with them.

数值算例表明，基于解析试函数的广义协调元具有良好的性能，计算精度高，并较好地解决了有限元发展中的几个疑难问题。

Via these formulas, the numerical solutions are incorporated with the analytical solution on the auxiliary circle.

在此基础上进一步建立了杂交元算式，它能保证在圆形人工边界上数值解和解析解两种场函数及其导数的相容性。

Moreover, the quasi-analytical expressions of the relative concentrations at the mobile region and the immobile region are deduced respectively by using Bessel equation, Bessel function and the method of Laplace transform under the initial concentration zero and the first or second boundary condition of a semi-infinite one-dimensional space.

并分别在初始浓度为零，半无限一维空间内第一类边界条件和第二类边界条件下，利用Bessel方程、Bessel函数和Laplace变换法推导出了可动区和不动区溶质相对浓度的准解析表达式。

The relation of displacement of tympanic membrane to sound pressure was educed by using the Bessel function; subsequently the analytic solution was obtained.

并利用贝赛尔函数建立耳膜位移与声压的关系，得到耳膜位移的解析解。

Taking beam with rectangular section as an example, the corresponding analytical solution of beam subjected to cosine distribution pressure is given through constructing the stress function satisfying all boundary conditions and biharmonic equation , which is a base of beam subjected to arbitrary distribution pressure.

以矩形截面简支梁为例，通过构造一个满足所有边界条件和双调和方程的应力函数，给出了梁受余弦分布力作用情况下相应的解析解，这为求解梁受任意分布压力作用下解的问题打下了基础

As she looked at Warrington's manly face, and dark, melancholy eyes, she had settled in her mind that he must have been the victim of an unhappy attachment.

每逢看到沃林顿那刚毅的脸，那乌黑、忧郁的眼睛，她便会相信，他一定作过不幸的爱情的受害者。

Maybe they'll disappear into a pothole.

也许他们将在壶穴里消失