级数展开

If the power series ∑anxn an ∑anxn and ∑bnxn satisfy the condition r =∑bn xn, we obtain the twin combinatorial identity theorem between the sequences {an} and bn}.We also obtain some specific twin combinatorial identities by using the expansion on the binomial expression formula, including two expansions of combinatorial number(rsn and ...

如果幂级数∑anxn与∑bnxn满足条件r＝∑bnxn 时，获得数列｛an｝与｛bn｝之间孪生组合恒等式的定理，应用在二项式定理等展开式上得出具体的多组孪生组合恒等式，其中包含组合数的两种展开法，Bernoulli数直接表达式的新证等结果。

Via the stochastic averaging method, the four-dimensional system is reduced to a two-dimensional one. Through the logarithmic polar transformation , and then Girsanov theorem and Feynmann-Kac formula, the backword differential operator is then obtained. By expanding the eigenfunctions as a Fourier cosine series, the approximate analytic expansion of the moment Lyapunov exponent is then obtained.

首先在经典二元机翼颤振方程加入随机激励，通过随机平均法，Girsanov定理和Feynmann-Kac公式得到关于矩Lyapunov指数的特征值问题，其次采用Fourier余弦级数对特征函数进行正交展开得到系统矩Lyapunov指数的近似解析式。

The distribution function of the system response and state function is approximately determined by the standard normal distribution functions using Edgeworth series, and its reliability is obtained.

在结构系统响应和状态函数的前四阶矩的一般数学表达式的基础上，使用 Edgeworth级数把未知响应和状态函数的概率分布展开成标准的正态分布表达式，并放松了对随机参数的分布概型和激励类型的限制，从而确定了系统的可靠度

The time-domain response of the induced currents on a scattering object irradiated by a plane wave with a Gaussian pulse in time is expanded as an associate Hermite series in this paper. Using the isomorphism of the associate Hermite function and its Fourier transform, the frequency-domain information can be obtained similarly to the time-domain expansion. According to the corresponding relation between time-domain analysis and frequency-domain analysis, and using the early-time response and the low-frequency information of the induced currents, the non-known coefficients in the expansion series can be decided. The entire responses of the induced currents on the scattering object in both time and frequency domains can be obtained at one time by simultaneous extrapolation computation using both the early-time response and the low-frequency information of the induced currents.

摘要首先根据散射体在高斯脉冲平面波激励下感应电流的能量几乎全部集中在时间轴和频率轴上的有限范围内，该文将时域响应展开为系数待定的连带Hermite级数的叠加，并根据连带Hermite函数的傅里叶变换的自反性，得到与时域响应形式类似的频域响应；然后利用时域方法和频域方法分别计算散射体上感应电流的早时响应和低频信息；最后经过时域和频域联合外推计算，由早时响应和低频信息确定时域和频域响应的待定系数，从而获得了整个时域和频域的完全响应。

By means of the expansion theorem, the displacement is expressed as the summation of the first several exact modal vectors.

考虑前若干阶线性振动模态，使用展开定理，将索的位移表示成共轭的模态函数的级数和。

"It is 我们想要 desired to develop this function into a Fourier series.

我们想要我们想要把这个函数展开成傅氏级数。

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。

It is widely used to develop complex functions into Laurent series at the neighborhood of a pole.

在数学中经常要用到复变函数在极点邻域内展开的罗朗级数。

For the compressible materials, in order to satisfy the higher-order compatibility equation of the rate of deformation in the centered fan sector, which is a quite difficult problem unresolved in the previous studies, a particular term must be added to the expansions of the stress components in the regular logarithmic power series adopted by many researchers before. A higher-order near tip field with tangential velocity jump ahead of the crack tip is derived first, of which the dominant terms are the solution widely accepted.

对可压缩材料，本文首先采用非规则的对数幂级数渐近展开方法，克服了过去研究中存在的中心扇形区变形率协调方程的高次渐近式不能被满足的困难，并得到了一个切向速度在裂尖前方存在间断的高阶渐近解，该解的主项就是目前被广泛接受的关于可压缩材料的尖端场解。

The detailed comparisons between the higher—order near tip solution and the well—known modified Prandtl field are also given in this paper.For the compressible materials, in order to satisfy the higher-order compatibility equation of the rate of deformation in the centered fan sector, which is a quite difficult problem unresolved in the previous studies, a particular term must be added to the expansions of the stress components in the regular logarithmic power series adopted by many researchers before. A higher-order near tip field with tangential velocity jump ahead of the crack tip is derived first, of which the dominant terms are the solution widely accepted.

对可压缩材料，本文首先采用非规则的对数幂级数渐近展开方法，克服了过去研究中存在的中心扇形区变形率协调方程的高次渐近式不能被满足的困难，并得到了一个切向速度在裂尖前方存在间断的高阶渐近解，该解的主项就是目前被广泛接受的关于可压缩材料的尖端场解。

However, as the name（read－only memory）implies, CD disks cannot be written onorchanged in any way.

然而，正如其名字所指出的那样，CD盘不能写，也不能用任何方式改变其内容。

Galvanizes steel pallet is mainly export which suits standard packing of European Union, the North America. galvanizes steel pallet is suitable to heavy rack. Pallet surface can design plate type, corrugated and the gap form, satisfies the different requirements.

镀锌钢托盘多用于出口，替代木托盘，免薰蒸，符合欧盟、北美各国对出口货物包装材料的法令要求；喷涂钢托盘适用于重载上货架之用，托盘表面根据需要制作成平板状、波纹状及间隔形式，满足不同的使用要求。