解析函数

The second kind of parabolic differential equation,multi-dimensional fractional equation,is generalized from the seepage flow.

遗憾的是，大多数分数阶微分方程的解析解都含有复杂的级数或者特殊函数，不利于进行近似计算。

As well known, modern laser diffraction has widely used in engineering measurement but it mostly limited in Fraunhofer diffraction because there is simple function between the size to be measured and space of diffraction dark spots and there are no analytic algorithm for Fresnel diffraction.

现代激光技术提供了极为良好的相干光源，该光源在工程测量，特别是衍射测量中得到广泛应用。但在实际应用中，往往局限于夫琅和费衍射，因为夫琅和费衍射法暗斑间距与被测尺寸之间具有很简单的函数关系S-n/d，而菲涅尔衍射光强分布非常复杂，缺乏解析的计算方法。

Based on the Fresnel diffraction integral and the complex Gaussian expansion for the hard-edge aperture function, a simple and analytical expression for calculating the diffraction field distribution of an apertured lens is presented.

基于菲涅尔衍射积分和将硬边孔径函数展开成有限项复高斯级数方法推导出了计算透镜衍射光强分布的解析表达式。

The existence and uniqueness of solution for FROE is proven,and its analytical solution is given.

证明了分数阶Relaxation-Oscillation方程解的存在惟一性，并利用格林函数给出了它的解析解。

Finally, based on complex function theory, we also gave the analysis solutions of this problem.

最后，根据复变函数理论求解范数的方法求得了问题的解析解。

The main results are as follows: the relations between local fractional integrated semigroups and the corresponding Cauchy problem, global fractional integrated semigroups and regularized semigroups are given; introduction of the notion of regularized resolvent families, and the generation theorem and analyticity criterions for regularized resolvent families are obtained; the spectral inclusions between fractional resolvent family and its generator, and the approximation for fractional resolvent families in the cases of generators approximation and fractional orders approximation; elliptic operators with variable coefficients generating fractional resolvent family on L^2 by using numerical range techniques; and the L^p theory for elliptic operators with real coefficients highest order are obtained by Sobolev''s inequalities and the a priori estimates for elliptic operators; and a kind of coercive differential operators generates fractional regularized resolvent family by applying the Fourier multiplier method, functional calculus and some basic properties of Mittag-Leffler functions.

主要结论是：给出了局部分数次积分半群和相应的Cauchy问题的关系以及分数次积分半群和正则半群的关系；引入了正则预解族的概念，并给出了其生成定理和解析生成法则；给出了分数次预解族与其生成元的谱包含关系，并研究了在生成元逼近和分数阶逼近两种情况下相应的预解族的逼近问题；利用数值域方法证明了具变系数的椭圆算子在L^2上生成分数次预解族；利用Sobolev不等式和椭圆算子的先验估计证明了具变系数的椭圆算子在其最高项系数为实数时在L^p上生成分数次预解族；运用Fourier乘子理论、泛函演算和Mittag-Leffler函数证明了一类强制微分算子可以生成分数次正则预解族，并给出了该预解族的范数估计。

In Chapter 1, concepts of iteration, dynamical system, iterative functional equation and iterative functional differential equation are introduced.

本文将研究几种类型的迭代函数方程和迭代泛函微分方程的光滑解和解析解的存在性、唯一性和稳定性。

In this paper existence of local analytic solutions of a polynomial-like iterative functional equation is studied.

本文讨论一类多项式型迭代函数方程局部解析解的存在性。

Using spheric functions and Galerkin's method, analytical solutions of the problems are obtained.

采用球函数和Galerkin方法给出了问题的解析解。

Based on the second-order moment of the power density, the far-field divergence angle of nonparaxial rotationally symmetric Laguerre-Gaussian beams is derived and expressed in a sum of the series of the Gamma function.

基于功率密度的二阶矩方法，推导出了非傍轴旋转对称拉盖尔－高斯光束远场发散角的解析公式，并表示为伽玛函数的幂级数求和形式。

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