A QUADRATURE FORMULA OF RATIONAL TYPE FOR INTEGRANDS WITH ONE ENDPOINT SINGULARITY

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Rational Solutions of the A_4^{(1)} Painlev'e Equation

$Rational Solutions of the A_4^{(1)} Painlev'e Equation$

In the third section, from the residue theorem, we get necessary condition for the parameters (αi)0≤i≤4 to have rational solutions. In addition, we transform the parameters to the fundamental domain of W˜ (A(41)); 0 ≤ αi ≤ 1 (0 ≤ i ≤ 4).
(αi, αi+1, αi+2, αi+3, αi+4) =
±
1 3
(1,
1,
1,
0,
0)
mod Z
±
1 3
(1,
−1,
−1,
1,
0)
mod Z.
(3) For some i = 0, 1, . . . , 4,
(αi, αi+1, αi+2, αi+3, αi+4) =
5j5j((11,,21,,11,,31,,31))
Introduction
In this paper, we will classify all of rational solutions of the A(41) Painlev´e equation. The A(41) Painlev´e equation is a member of the family of the A(n1) Painlev´e equations found by Noumi-Yamada [7]. The A(21) Painlev´e equation corresponds to the fourth Painlev´e equation. The A(41) Painlev´e equation is the ﬁrst equation of the A(n1) Painlev´e equations, which are not the original Painlev´e equations.

美联英语：数学词汇

美联英语提供：美联英语：数学词汇小编给你一个美联英语官方试听课申请链接：/?tid=16-73374-0 Mathematics弧arc弦chord实数real number虚数imaginary number整数integral number分数fraction number奇数odd number偶数even number正数positive number负数negative number基数cardinal number序数ordinal number加法addition加号plus加数adbent减法subtraction减号minus减数subtrahend乘法multiplication乘号multiplication 两倍duple三倍triple四倍quadruple五倍quintuple六倍sextuple七倍sevenfold八倍eightfold九倍ninefold十倍tenfold百倍centuple乘数multiplier除法division除号division除数divisor商数quotient分子numerator分母denominator 小数decimal系数coefficient因数factor函数function对数logarithm常数constant代数algebra积分integration 微分differential比例proportion 序列sequence排列permutation 求积quadrature 乘方involution指数index支点branch point 切线tangent line 曲线curve共线collinearity 实线full line虚线dotted line 直线right line同轴coaxial纵轴axis ordinate 横轴axis abscissae 切面tangent plane锥面conical face曲面curved face侧面lateral face梯形trapezoid圆形circle中心center圆周circumference 椭圆ellipse弧度circular measure 半圆semicircle半径radius直径diameter切圆circle contact对称symmetry对边opposite side 斜边hypotenuse夹边contained side 斜高slant height面积area角距argular distance 对角opposite angle 钝角obtuse angle锐角acute angle夹角contained angle定理proposition公式formula补角complemenary angle内角interior angle外角exterior angle有理数rational number无理数irrational number自然数natural number被减数minuend被乘数multiplicand被除数dividend小数点point百分比percent已知数datum近似数approximate number相关数correlative绝对数absolute number循环数period代数和algebraic sum开平方extraction of square root平方根square root立方根cube root无穷小infinitely small无穷大infinitely great不等式inequality方程式equation平分线bisector水平面horizontal plane五角形pentagon六角形hexagon七角形heptagon八角形octagon对角线diagonal正方形square长方形rectangle立方体cube球面圆circle of sphere不对称asymmetry同位角correspinding angle九九乘表multiplication table近似等值approximate equivalent 常用对数common logarithm代数运算algebraic operation垂直平面vertical plane平面三角plane griangle等边三角equilateral triangle钝角三角obtuse triangle一次方程式system of linear equation 二次方程式affected quadratic equation 三次方程式affected cubic equation四次方程式biquadratic equation一元方程式eqution with one unknown 二元方程式eqution with two unknown 三元方程式equatic with three unknown 平行四边形parallelongram直角三角形right triangle美联英语：。

电子信息类专业英语(西电第二版)Unit 4 Microel

Unit 4 Microelectronics
Integrated circuits are small, light, rugged, and reliable. They require less power and lower voltages than equivalent macroscopic circuits; consequently they operate at lower temperatures, and individual components may be close together without exceeding the operating temperature limit. Relatively little stray capacitance and short time delays are produced because of the short interconnections between the individual components in IC. Maintenance is simplified because if a component of the IC fails the complete IC is usually replaced. Mass production techniques of plane technology have reduced the cost of many IC so that they are almost as inexpensive as a single transistor. Eventually most conventional circuits will be replaced by IC.
［2］ The reduction in power dissipation is particularly important where a complex circuit is to be concentrated into a small space. The problem of extracting the heat generated in the circuit may then be a difficult one.

数值分析中常用数学词汇英中文对照翻译.

数值分析中常用数学词汇英中文对照abbreviation 简写符号;简写absolute error 绝对误差absolute value 绝对值accelerate 加速accumulation 累积accuracy 准确度act on 施于action 作用; 作用力add 加addition 加法addition formula 加法公式addition law 加法定律additive property 可加性adjoint matrix 伴随矩阵algebra 代数algebraic 代数的algebraic equation 代数方程algebraic expression 代数式algebraic fraction 代数分式;代数分数式algebraic inequality 代数不等式algebraic number 代数数algebraic operation 代数运算algorithm 算法系统; 规则系统alternating series 交错级数alternative hypothesis 择一假设; 备择假设; 另一假设analysis 分析;解析angle 角anti-clockwise direction 逆时针方向;返时针方向anti-derivative 反导数; 反微商anti-logarithm 逆对数;反对数anti-symmetric 反对称approach 接近;趋近approximate value 近似值approximation 近似;略计;逼近Arabic system 阿刺伯数字系统arbitrary 任意arbitrary constant 任意常数arc 弧arc-cosine function 反余弦函数arc-sin function 反正弦函数arc-tangent function 反正切函数area 面积argument (1论证; (2辐角argument of a function 函数的自变量arithmetic 算术array 数组; 数组ascending order 递升序ascending powers of X X 的升幂assumption 假定;假设asymmetrical 非对称asymptote 渐近augmented matrix 增广矩阵average 平均;平均数;平均值axiom 公理back substitution 回代base (1底;(2基;基数basis 基belong to 属于bias 偏差;偏倚billion 十亿binary number 二进数binary operation 二元运算binary system 二进制binomial 二项式bisection method 分半法;分半方法boundary condition 边界条件boundary line 界(线;边界bounded 有界的bounded above 有上界的;上有界的bounded below 有下界的;下有界的bounded function 有界函数bounded sequence 有界序列brace 大括号bracket 括号breadth 阔度calculation 计算calculator 计算器;计算器calculus (1 微积分学; (2 演算cancel 消法;相消Cartesian coordinates 笛卡儿坐标category 类型;范畴centre 中心;心chain rule 链式法则chance 机会change of base 基的变换change of variable 换元;变量的换characteristic equation 特征(征方程characteristic function 特征(征函数characteristic root 特征(征根chart 图;图表check digit 检验数位checking 验算circle 圆classification 分类clockwise direction 顺时针方向clockwise moment 顺时针力矩closed convex region 闭凸区域closed interval 闭区间coefficient 系数cofactor 余因子; 余因式coincide 迭合;重合collection of terms 并项collinear 共线collinear planes 共线面column (1列;纵行;(2 柱column matrix 列矩阵column vector 列向量combination 组合common denominator 同分母;公分母common difference 公差common divisor 公约数;公约common factor 公因子;公因子common multiple 公位数;公倍comparable 可比较的complement 余;补余completing the square 配方complex number 复数complex number plane 复数平面complex root 复数根component 分量composite function 复合函数; 合成函数computation 计算computer 计算机;电子计算器concept 概念conclusion 结论condition 条件conditional 条件句;条件式conjugate 共轭constant 常数constant of integration 积分常数constraint 约束;约束条件continuity 连续性continuous function 连续函数contradiction 矛盾converge 收敛convergence 收敛性convergent 收敛的convergent iteration 收敛的迭代convergent sequence 收敛序列convergent series 收敛级数convex 凸convexity 凸性coordinate 坐标corollary 系定理; 系; 推论correspondence 对应counter clockwise direction 逆时针方向;返时针方向counter example 反例counting 数数;计数criterion 准则critical point 临界点critical region 临界域cirtical value 临界值cube 正方体;立方;立方体cubic 三次方;立方;三次(的 cubic equation 三次方程cubic roots of unity 单位的立方根cumulative 累积的curve 曲线decimal 小数decimal place 小数位decimal point 小数点decimal system 十进制definite integral 定积分definition 定义degree (1 度; (2 次degree of a polynomial 多项式的次数degree of accuracy 准确度degree of ODE 常微分方程次数delete 删除; 删去denary number 十进数denominator 分母dependence (1相关; (2应变derivable 可导derivative 导数determinant 行列式diagonal 对角线diagonal matrix 对角矩阵difference 差difference equation 差分方程differentiable 可微differential 微分differential coefficient 微商; 微分系数differential equation 微分方程differential mean value theorem 微分中值定理differentiate 求...的导数differentiation 微分法digit 数字dimension 量; 量网; 维(数direction 方向; 方位discontinuity 不连续性discontinuous 间断(的;连续(的; 不连续(的discontinuous point 不连续点discrete 分立; 离散distance 距离diverge 发散divergence 发散(性divergent 发散的divergent iteration 发散性迭代divergent sequence 发散序列divide 除dividend (1被除数;divisible 可整除division 除法division algorithm 除法算式divisor 除数;除式;因子dot 点dot product 点积echelon form 梯阵式echelon matrix 梯矩阵eigenvalue 本征值eigenvector 本征向量element 元素elementary row operation 基本行运算elimination 消法elimination method 消去法;消元法empty set 空集equivalent 等价(的error 误差error estimate 误差估计error term 误差项estimate 估计;估计量evaluate 计值exact 真确exact solution 准确解;精确解;真确解exact value 法确解;精确解;真确解example 例expand 展开experiment 实验;试验experimental 试验的exponent 指数exponential function 指数函数express…in terms of… 以………表达extreme point 极值点extreme value 极值extremum 极值factor 因子;因式;商factor method 因式分解法factorial 阶乘factorization 因子分解;因式分解fallacy 谬误FALSE 假(的falsehood 假值finite 有限finite sequence 有限序列first derivative 一阶导数first order differential equation 一阶微分方程fixed point 不动点fixed point iteration method 不动点迭代法for all X 对所有X for each /every X 对每一Xform 形式;型format 格式;规格formula(formulae 公式fraction 分数;分式function 函数fundamental theorem of calculus 微积分基本定理Gaussian elimination 高斯消去法general form 一般式;通式general solution 通解;一般解general term 通项given 给定;已知global 全局; 整体global maximum 全局极大值; 整体极大值global minimum 全局极小值; 整体极小值gradient (1斜率;倾斜率;(2梯度graph 图像;图形;图表graphical method 图解法graphical representation 图示;以图样表达graphical solution 图解growth 增长higher order derivative 高阶导数horizontal 水平的;水平hypothesis 假设identity 等(式identity matrix 恒等矩阵if and only if/iff 当且仅当;若且仅若if…, then 若….则;如果…..则illustration 例证;说明image 像点;像imaginary number 虚数implicit function 隐函数imply 蕴涵;蕴含improper integral 广义积分; 非正常积分increase 递增;增加indefinite integral 不定积分independence 独立;自变inequality 不等式;不等inequality sign 不等号infinite 无限;无穷infinite sequence 无限序列;无穷序列infinite series 无限级数;无穷级数infinitesimal 无限小;无穷小infinity 无限(大;无穷(大initial approximation 初始近似值initial condition 原始条件;初值条件initial value 初值;始值initial-value problem 初值问题inner product 内积input 输入integer 整数integral 积分integrate 积;积分;......的积分integration 积分法integration by parts 分部积分法integration by substitution 代换积分法;换元积分法interchange 互换intermediate value theorem 介值定理interpolating polynomial 插值多项式interpolation 插值interval 区间intuition 直观invalid 失效;无效invariance 不变性invariant (1不变的;(2不变量;不变式inverse 反的;逆的inverse function 反函数;逆函数inverse matrix 逆矩阵inverse problem 逆算问题invertible 可逆的invertible matrix 可逆矩阵iterate (1迭代值; (2迭代iteration 迭代iterative method 迭代法known 己知Lagrange interpolating polynomial 拉格朗日插值多项代leading coefficient 首项系数leading diagonal 主对角线lemma 引理limit 极限limit of sequence 序列的极限line of best-fit 最佳拟合line segment 线段linear 线性;一次linear convergence 线性收敛性linear differeantial equation 线性微分方程linear equation 线性方程;一次方程linear equation in two unknowns 二元一次方程;二元线性方程linearly dependent 线性相关的linearly independent 线性无关的local maximum 局部极大(值local minimum 局部极小(值logic 逻辑long division method 长除法loop 回路lower bound 下界lower triangular matrix 下三角形矩阵Maclaurin expansion 麦克劳林展开式magnitude 量;数量;长度;大小mantissa 尾数matrix 阵; 矩阵matrix addition 矩阵加法matrix equation 矩阵方程matrix multiplication 矩阵乘法matrix operation 矩阵运算maximize 极大maximum absolute error 最大绝对误差mean value theorem 中值定理method of completing square 配方法method of interpolation 插值法; 内插法method of least squares 最小二乘法; 最小平方法method of substitution 代换法;换元法method of successive substitution 逐次代换法; 逐次调替法minimize 极小minus 减modulus of a complex number 复数的模monomial 单项式multiple 倍数multiple root 多重根multiplication 乘法multiplicity 重数multiplier 乘数;乘式multiply 乘mutually independent 独立; 互相独立mutually perpendicular lines 互相垂直n factorial n阶乘n th derivative n阶导数n th root n次根;n次方根n the root of unity 单位的n次根natural logarithm 自然对数necessary and sufficient condition 充要条件necessary condition 必要条件negative 负neighborhood 邻域Newton-Cote's rule 牛顿- 高斯法则Newton-Raphson's method 牛顿- 纳逊方法Newton's formula 牛顿公式Newton's method 牛顿方法non-linear 非线性non-linear equation 非线性方程non-negative 非负的non-singular (1满秩的; (2非奇异的non-singular matrix 满秩矩阵non-trivial 非平凡的non-zero 非零norm 模方; 范数normal (1垂直的;正交的;法线的(2正态的(3正常的;正规的normalize 正规化normalized form 标准型notation 记法;记号null 零; 空null set 空集null vector 零向量number 数numerator 分子numerical method 计算方法;数值法objective function 目标函数octant 卦限odd function 奇函数one-to-one 一个对一个one-one correspondence 一一对应operation 运算order of a matrix 矩阵的阶ordinary differential equation 常微分方程origin 原点orthogonal 正交orthogonality 正交性 outcome 结果 output 输出 parameter 参数；参变量parametric equation 参数方程 partition 分割; 划分 periodic function 周期函数permutation 排列 perpendicular 垂线；垂直(于 phase 相; 位相 pivot 支点 plot 绘图plus 加 point 点 polynomial 多项式 polynomial equation 多项式方程 positive 正 post-multiply 后乘; 自右乘 premultiply 前乘; 自左乘 prime 素 product 乘积；积 proper integral 正常积分 property 性质 quadratic convergence 二阶收敛性 quadratic formula 二次公式 quadratic function 二次函数 quadratic inequality 二次不等式 quadrature 求积法 quadrilateral 四边形 quotient 商；商式 quotient rule 商法则 R.H.S 右 rank 秩 rate of convergence 收敛率 ratio 比 ; 比率 rational function 有理函数 real number 实数 real part 实部 real root 实根 reciprocal 倒数 rectangle 长方形；矩形 recurrence formula 递推公式 recurrent 循环的 recurring decimal 循环小数 reduce 简化 region 区域 region of convergency 收敛区域 regular 正；规则 relative error 相对误差 remainder term 余项root 根 rotation 旋转 rounded number 舍数 rounding(off 舍入；四舍五入 row 行；棋行 row vector 行向量; 行矢量 rule 规则；法(则 satisfy 满足；适合 scalar 纯量; 无向量, 标量 scalar matrix 纯量矩阵 scale 比例尺；标度；图尺 scientific notation 科学记数法 secant (1正割; (2割线 secant method 正割法 second derivative 二阶导数 second order ordinary differential equation 二阶常微分方程 sentence 句；语句 sequence 序列series 级数 set 集 shaded portion 有阴影部分 shape 形状 shear 位移 side 边；侧 sign 符号；记号 signed number 有符号数 significant figure 有效数字 signum 正负号函数similar 相似 simplify 简化 Simpson's integral 森逊积分 Simpson's rule 森逊法则singular 奇的 singular matrix 奇异矩阵; 不可逆矩阵 span 生成 square (1平方；(2正方形 square bracket 方括号square matrix 方(矩阵 stability 稳度 stationary 平稳 stationary point 平稳点; 逗留点; 驻点 straight line 直线 subset 子集 substitute 代入 substitution 代入; 代入法subtract 减 subtraction 减法 successive approximation 逐次逼近法 successive derivative 逐次导数 successive differentiation 逐次微分法 sufficiency 充份性 sufficient and necessary condition 充要条件 sufficient condition 充份条件 sufficiently close 充份接近suffix 下标 sum 和 summation 求和法; 总和 symbol 符号; 记号 symmetry 对称; 对称性Taylor’s expansion 泰勒展开式 term 项 transpose 移项；转置 transpose of matrix 倒置矩阵；转置矩阵 trapezium 梯形 trapezoidal integral 梯形积分 trapezoidal rule 梯形法则 trial 试；试验 triangle 三角形 triangular matrix 三角矩阵 trigonometric equation 三角方程 trigonometric function 三角函数 triple 三倍 trivial solution 平凡解truncation error 截断误差 undefined 未下定义(的 undetermined coefficient 待定系数unequal 不等 unique solution 唯一解 uniqueness 唯一性 unit 单位 unit area 单位面积unit circle 单位圆 unknown 未知数；未知量 upper bound 上界 upper limit 上限 upper triangular matrix 上三角形矩阵 validity 真确性; 有效性 variable 变项；变量；元；变元；变数 vector 向量; 矢量 vector function 向量函数; 矢量函数 vector product 矢量积; 矢量积 vector space 向量空间 verify 证明；验证 weight (1重量；(2权 weighted average, weighted mean 加权平均数 without loss of generality 不失一般性 x-axis x 轴x-coordinate x 坐标 x-intercept x 轴截距 y-axis y 轴 y-coordinate y 坐标 y-intercept y轴截距 zero 零 zero factor 零因子 zero matrix 零矩阵 zero vector 零向量 zeros of a function 函数零值。

Singularities of Rational Curves on K3 surfaces

Date : February 1, 2008.
1
2
XI CHEN
over ∆, whose general ﬁber Υt is a rational curve in the linear series |OXt (1)| for each t, the central ﬁber Υ0 will be a curve in the linear series |OR (1)| and hence Υ0 = Σ1 ∪ Σ2 where Σi is a curve of type (1, k ) on Ri . Our trivial observation is that Υt is nodal if Υ0 is nodal. However, Υ0 could fail to be nodal where 1. it has a reduced (i.e. isolated) singularity other than a node; or 2. it is nonreduced. The ﬁrst case turns out much easier to handle than the second. This is basically due to the fact that each Σi is a curve of type (1, k ) on P1 × P1 . So all the isolated singularities of Σi are nodes. If Υ0 = Σ1 ∪ Σ2 has an isolated singularity other than a node, it must be one of the intersections between Σ1 and Σ2 on E . The deformation of such singularities has been studied in [C]. With a bit more care, we are able to show that these singularities deform to nodes on the general ﬁber Υt . However, if Υt is a rational curve in a multiple of the primitive class, Σi might have isolated singularities other than nodes which have to be taken care of. This is one of the major obstacles to generalize Theorem 1.1 to all rational curves on K3 surfaces. To handle the second case, i.e., to handle the nonreduced components of Υ0 , we ﬁrst divide them into three types, which we will call Type I, II or III chain (see Sec. 2), respectively. The deformation of Υ0 along a Type I chain is studied in Sec. 3. The basic technique used there is to normalize the total family along the Type I chain after a suitable base change. The deformation of Υ0 along a Type II chain is studied in Sec. 4, where we build our argument upon a lower bound estimation on the δ -invariant of Υt in the neighborhood of a Type II chain. The deformation of Υ0 along a Type III chain is studied in Sec. 5. This turns out to be the hardest case among the three. A two-stage degeneration is used, First, we degenerate a general K3 surface to an elliptic K3 (see Sec. 5); and then we degenerate an elliptic K3 to the union of two rational surfaces described above. The degeneration of a K3 surface to an elliptic K3 is also an important step in Bryan and Leung’s work [B-L], although the elliptic K3 surfaces they used are diﬀerent from the ones we use. As a side note, there have been several progresses made on the enumeration problems on K3 surfaces following Yau and Zaslow’s work. A. Beauville pointed out that the numbers Yau and Zaslow obtained are the numbers of rational curves in |OS (1)| with each curve counted with certain multiplicity [B], the multiplicity of a rational curve only depends on its singularities and is 1 if the curve is nodal. He gave an algebraic deﬁnition of the multiplicy. Later B. Fantechi, L. G¨ ottsche

卡瑞尔公式英文

卡瑞尔公式英文The Carrel Formula is a mathematical formula developed by Dr. Alexis Carrel, a French surgeon and biologist, in the early 20th century. This formula is used to estimate the required number of cells for the growth of tissues in culture, specifically for maintaining the vitality of organ tissues outside the living body. The Carrel Formula is an important tool in the field of tissue culture and has been widely used in research and regenerative medicine.The formula is given as follows:N = V x P x FWhere:N: The number of cells required for tissue cultureV: The volume of the culture mediumP: The proliferation rate of the cellsF: The fraction of the tissue made up of cellsThe Carrel Formula takes into account the key factors that influence the growth and maintenance of tissue cultures. The volume of the culture medium is an important parameter as it provides the necessary nutrients and growth factors for the cells. The larger the volume, the more cells are required to maintain tissue vitality.The proliferation rate of the cells is another crucial factor. This refers to the rate at which the cells divide and multiply. Different cell types have different proliferation rates. Some cells may dividerapidly, while others may have a slower rate of division. It is important to determine the specific proliferation rate for a given cell type in order to accurately calculate the number of cells needed for tissue culture.The fraction of the tissue made up of cells is also considered in the Carrel Formula. Tissues are composed of cells as well as extracellular matrix and other components. Depending on the type of tissue, the fraction of cells can vary. For example, highly cellular tissues such as blood or epithelial tissues have a higher fraction of cells compared to tissues with dense extracellular matrix such as cartilage. This factor ensures that the appropriate number of cells is provided to maintain tissue integrity.The Carrel Formula is a useful guideline for researchers and clinicians working with tissue cultures. By estimating the number of cells required, it helps in planning experiments and determining the amount of resources needed. However, it's important to note that the formula provides an estimate and may need to be adjusted based on specific experimental conditions and variations observed in different cell lines or tissues.In conclusion, the Carrel Formula developed by Dr. Alexis Carrel provides a mathematical approach to estimate the number of cells required for tissue culture. This formula takes into account the volume of the culture medium, the proliferation rate of the cells, and the fraction of the tissue composed of cells. It is a valuable tool in tissue engineering and regenerative medicine, aiding in the design and optimization of experiments involving tissue cultures.。

数学英语词汇

数学英语词汇数学英语词汇表[翻译必看]一般词汇数学mathematics, maths(BrE), math(AmE)公义axiom定理theorem计算calculation运算operation证明prove假定hypothesis, hypotheses(pl.)命题proposition算术arithmetic加plus(prep.), add(v.), addition(n.)被加数augend, summand加数addend和sum减minus(prep.), subtract(v.), subtraction(n.)被减数minuend减数subtrahend差remainder乘times(prep.), multiply(v.), multiplication(n.)被乘数multiplicand, faciend乘数multiplicator积product除divided by(prep.), divide(v.), division(n.)被除数dividend除数divisor商quotient等于equals, is equal to, is equivalent to大于is greater than小于is lesser than大于等于is equal or greater than小于等于is equal or lesser than运算符operator数字digit数number自然数natural number数学英语词汇整数integer小数decimal小数点decimal point分数fraction分子numerator分母denominator比ratio正positive数学英语词汇负negative零null, zero, nought, nil十进制decimal system二进制binary system十六进制hexadecimal system权weight, significance进位carry截尾truncation四舍五入round下舍入round down上舍入round up有效数字significant digit无效数字insignificant digit代数algebra公式formula, formulae(pl.)单项式monomial多项式polynomial, multinomial系数coefficient未知数unknown, x-factor, y-factor, z-factor等式，方程式equation一次方程simple equation二次方程quadratic equation三次方程cubic equation四次方程quartic equation不等式inequation阶乘factorial对数logarithm指数，幂exponent乘方power二次方，平方square三次方，立方cube四次方the power of four, the fourth powern次方the power of n, the nth power开方evolution, extraction二次方根，平方根square root三次方根，立方根cube root数学英语词汇四次方根the root of four, the fourth rootn次方根the root of n, the nth root会合aggregate元素element空集void子集subset交集intersection并集union补集complement映照mapping函数function定义域domain, field of definition 值域range常量constant变量variable单一性monotonicity奇偶性parity周期性periodicity图象image数列，级数series微积分calculus微分differential导数derivative极限limit无量大infinite(a.) infinity(n.) 无量小infinitesimal积分integral定积分definite integral不定积分indefinite integral有理数rational number无理数irrational number实数real number虚数imaginary number复数complex number矩阵matrix队列式determinant几何geometry点point线line面plane体solid线段segment射线radial平行parallel订交intersect角angle角度degree弧度radian锐角acute angle 直角right angle 钝角obtuse angle 平角straight angle数学英语词汇周角perigon底base边side高height三角形triangle锐角三角形acute triangle直角三角形right triangle直角边leg斜边hypotenuse勾股定理Pythagorean theorem钝角三角形obtuse triangle不等边三角形scalene triangle等腰三角形isosceles triangle等边三角形equilateral triangle四边形quadrilateral平行四边形parallelogram矩形rectangle长length宽width菱形rhomb, rhombus, rhombi(pl.), diamond正方形square梯形trapezoid直角梯形right trapezoid等腰梯形isosceles trapezoid五边形pentagon六边形hexagon七边形heptagon八边形octagon九边形enneagon十边形decagon十一边形hendecagon十二边形dodecagon多边形polygon正多边形equilateral polygon圆circle圆心centre(BrE), center(AmE)数学英语词汇半径radius直径diameter圆周率pi弧arc半圆semicircle扇形sector环ring椭圆ellipse数学英语词汇圆周circumference周长perimeter面积area轨迹locus, loca(pl.)相像similar全等congruent四周体tetrahedron五面体pentahedron六面体hexahedron平行六面体parallelepiped立方体cube七面体heptahedron八面体octahedron九面体enneahedron十面体decahedron十一面体hendecahedron十二面体dodecahedron二十面体icosahedron多面体polyhedron棱锥pyramid棱柱prism棱台frustum of a prism旋转rotation轴axis圆锥cone圆柱cylinder圆台frustum of a cone球sphere半球hemisphere底面undersurface表面积surface area体积volume空间space坐标系coordinates坐标轴x-axis, y-axis, z-axis横坐标x-coordinate数学英语词汇纵坐标y-coordinate原点origin双曲线hyperbola抛物线parabola三角trigonometry正弦sine余弦cosine正切tangent数学英语词汇余切cotangent正割secant余割cosecant反正弦arc sine反余弦arc cosine反正切arc tangent反余切arc cotangent反正割arc secant反余割arc cosecant相位phase周期period振幅amplitude心里incentre(BrE), incenter(AmE)外心excentre(BrE), excenter(AmE)旁心escentre(BrE), escenter(AmE)垂心orthocentre(BrE), orthocenter(AmE)重心barycentre(BrE), barycenter(AmE)内切圆inscribed circle外切圆circumcircle统计statistics均匀数average加权均匀数weighted average方差variance标准差root-mean-square deviation, standard deviation 比率propotion百分比percent百分点percentage百分位数percentile摆列permutation组合combination概率，或然率probability散布distribution正态散布normal distribution非正态散布abnormal distribution图表graph条形统计图bar graph柱形统计图histogram折线统计图broken line graph 曲线统计图curve diagram扇形统计图pie diagramwave equation颠簸方程tangental equation切线方程quartic equation四次方程式quadratic equation二次方程式magnitude equation【天】星等差linearequation一次方程式irreducibleequation不行约方程integralequation积分方程indicialequation【数】指数方程indeterminateequation不定方程数学专业英语Lesson 2Mathematics as a Language of Scienceassert vt.断言；坚持主张；保护表示qualitative adj.性质的；定性的quantitative adj.量的；数目的；定量的；与数目有关的astronomy n.天文学postulaten.假定,基本条件,基来源理vt.要求,假定vi.要求hypotheticaladj.假定的,假定的,爱猜想的deductionn.减除,扣除,减除额,推论,演绎induction n.概括；概括法；概括所得之结论verification n.考证；证明correlatevt.使互相关系vi.和...有关discardvt.扔掉,扔掉v.放弃discreditn.不相信；失期consistentadj.一致的,调解的,牢固的, [数、统]相容的inadequacyn.不充足,不合适,不合适,不足额conic, conicaladj圆锥的；圆锥形的ellipse n.椭圆,椭圆形ellipt (n.)hyperbolic adj.双曲线的hyperbola (n.)parabolic adj.用寓言表达的:抛物线的，像抛物线的parabola (n.) algebraic adj.代数的,对于代数学的mineralogy n.矿物学refractionn.折光,折射stimulusn.刺激物,促使要素,刺激,刺激impetusn.冲力推进力；刺激Lesson 3Axioms, definitions and Theoremsaxiomn. [数]公义definitionn.说明；确立定义；界说extravagantadj.豪侈的,浪费的,过分的,放纵的collinearadj.在同向来线上的,同线的convexadj.凸出的；凸面的segmentn.部分；片段；节,弓形；圆缺；弧形,线段conswquentlyadv.进而,所以in terms ofadv.依据,依照,用...的话,在...方面pretensen.主张,要求,伪称,借口,自称Lesson 4Geometry and Geometrical termstermn.学期,限期,期间,条款,条件,术语trianglen. [数]三角形,三人一组,三角关系parallelogramn.平行四边形straight anglen. [数]平角right anglen.直角acute anglen.锐角obtuse anglen.钝角reflex anglen.优角rectilinearadj直线的；由直线构成的；循直线进行的isosceles trianglen.等腰三角形equilateral trianglen.等边三角形right trianglen.直角三角形obtuse trianglen.钝角三角形acute trianglen.锐角三角形equiangular trianglen.正三角形,等角三角形hypotenusen.（直角三角形的）斜边circle圆center中心；中央；圆心diametern.直径radiusn.半径,范围,辐射光芒,有效航程,范围,界线circumferencen.圆周,四周Lesson 5The Method of Limitslimitn.限度,极限,极点infiniteadj.无穷的；无量的infinitesimaladj.无量小的,极小的,无穷小的calculusn.微积分学,结石exemplifyvt.例证,例示,作为...例子inscribev.记下polygonn. [数]多角形,多边形diminishv. (使)减少, (使)变小curvilinearadj曲线的,由曲线构成的intuitionn.直觉,直觉的知识integraln. [数学]积分,完好,部分defectiveadj.出缺点的, (智商或行为有)短缺的differential coefficient微分系数arithmeticaladj.算术的,算术上的convergencen.集中,收敛criterionn. (责备判断的)标准,准据,规范sequencen.序次,次序,序列irrational numbersn. [数]无理数domain，定义域contradiction矛盾reversaln.颠倒,反转,反向,逆转,撤除Lesson 6Functioncontinuous variable连续变量;［连续变数］variation变分,变化interval区间independent variable自变量dependent variable应变量rectangular coordinate直角坐标abscissan.〈数〉横坐标ordinaten. [数]纵线,纵座标gradientadj.倾斜的n.梯度,倾斜度,坡度slopen.斜坡,斜面,倾斜v. (使)顺斜Lesson 7Differential and Integral calculus differentialadj.微分的n.微分(differentiation) Integraln. [数学]积分,完好,部分(integration) calculusn.微积分学,结石interrelationn.互相关系trigonometryn.三角法exponentialadj.指数的,幂数的logarithmn. [数]对数derivativen.导数；微商tangentn.切线, [数]正切counterclockwiseadj.反时针方向的adv.反时针方向(clockwise) definite integral定积分approximationn.靠近,走近, [数]近似值culminatev.达到极点meann.均匀数,中间,中庸differentialequation微分方程extreme valuen.极值multiple integral多重积分double integralline integralfunctional analysis泛函剖析Lesson 8 The Concept of Cardinal Number (I) cardinal numbern.基数（如：1, 2, 3, ...有别于序数）denumerableadj.可数的aggregaten.共计,总计,会合体adj.共计的,会合的,聚合的v.齐集,会合,共计purportn.要旨v.宣称fanciern.幻想家,培养动物(或植物)的专家,喜好者sniffv.使劲吸,嗅,闻到,觉察,小看,使劲吸气n.吸,闻,吸气声,不屑一顾schemen.安排,配置,计划,阴谋,方案,图解,纲要v.计划,设计,图谋,策划, * n.（计算数学）方法，格式superiorn.长辈,能手,上司adj.较高的,上司的,上好的,出众的,傲慢的cumbersomeadj.厌烦的,麻烦的,粗笨的instructionn.指示,用法说明(书),教育,指导,指令drasticallyadv.强烈地,完全地conservation守衡律quadraturen.求积,求积分interpolationn.插值extrapolationn. [数]外推法,推测internal point内点identicaladj.同一的,相同的generalized solution广义解functional泛函hydrodynamics流体力学，水动力学divergence发散（性），梯度，发散量play an important (fundamental ... ) role起侧重要的（...）作用integro-interpolation method积分插值法Variational method变分方法comparativelyadv.比较地,相当地deficiencyn.缺少,不足fictiveadj.虚假的,想象上的,虚伪的self-adjoint (nonself-adjoint)自治的，自伴的，自共轭的finite element method有限元法spline approximation样条迫近Particles-in-the-Cell网格质点法heraldn.使者,传令官,通告者,前驱,先兆vt.预告,宣告,传达,喝彩advectionn.水平对流phenomenologicaladj.现象学的,现象的fluctuationn.颠簸,起伏optimismn.乐观,乐观主义pessimismn.消极,消极主义unjustifiedadj未被证明其正确的mean-square均方dispersionn. [数]离差,差量Polynomialn adj. [数]多项式的interpolation插值arithmeticn.算术,算法rounding errors舍入偏差multiplen.倍数,若干subjectiveadj.主观的,个人的objectiveadj.客观的,outcomen.结果,成就patternn.样品tossv.投,掷exhaustvt.用尽,耗尽,抽完,使筋疲力尽divisibleadj.可分的dice, dien.骰子assignvt.分派,指派attachvt.缚上,系上,贴上v.配属,隶属于pitfalln.缺点chairperson主席mechanicsn. (用作单数)机械学、力学, (用作复数)技巧,结构staticsn. [物]静力学dynamicsn.动力学adequatelyadv.充足地celestialadj.天上的macroscopicadj.肉眼可见的,巨观的classical field theory经典场理论rigitadj.刚硬的,刚性的,严格的elasticadj.弹性的plasticn.可塑的,塑性的,塑料的quantumn.量,额, [物]量子,量子论inceptionn.开初,获取学位pertainv.合适,属于gravitationn.地心吸力,引力作用tiden.潮,潮汐,潮流,趋向monumentaladj.纪念碑的,纪念物的,不朽的,特别的encompassv.包围,围绕,包含或包含某事物ingredientn.成分,要素acquaintedadj.有知识的,了解的synonymousadj.同义的configurationn.结构,结构,配置,外形referencen.说起,波及,参照,参照书目inertian.惯性,惯量attribute特征momentumn.动量proportionaladj.比率的,成比率的,相当的,均衡的designate指明negligibleadj.能够忽视的,不予重视的projectilen.射弹adj.发射的ballisticsn.弹道学,发射学intractableadj.难办理的{Mechanics of a Particlein consequence ofadv.因为的...缘由exertvt.尽(力),施加(压力等),努力v.发挥,全力以赴,尽galaxyn.星系,银河,一群显赫的人,一系列光彩醒目的东furnishvt.供应,供应,装备,部署v.供应torquen.扭矩,转矩moment力矩的friction摩擦dissipationn.消失,分别,挥霍,浪费,消遣,放纵,痛饮inferv.推测Hooke s Law and Its Consequenceselasticityn.弹力,弹性constitutiveadj.构成的,拟订的atomisticadj.原子论的crackn.裂痕,噼啪声v. (使)破碎,裂纹, (使)爆裂continuum mechanicsn.连续介质力学数学英语词汇superpositionn.重叠,重合,叠合strainn.过分的疲惫,紧张,张力,应变vt.扭伤,损害v.拉紧,扯紧, (使)紧张,全力thermodynamicsn. [物]热力学reckonvt.计算,总计,预计,猜想vi.数,计算,预计,依靠,猜想lesson 20strength强度load载荷empirical以经验为依照的member构件isolated孤立的segment部分、段、节stress应力strain应变tension拉伸shear剪切bend曲折torsion扭转、扭力insofar在⋯⋯范围cohesive内聚性的tensile拉力、张力stiffness硬度furnish供应Lesson 23 Fluid Mechanicseruption喷发、迸发turbulent湍流laminar层流isothermal等温isotropic各向同性prevalent广泛的、流行的tornado旋风、飓风数学英语词汇eddy旋涡viscosity粘性、粘度nonviscous无粘性的rotation旋转adiabatic绝热的reversible可逆的数学英语词汇isentropic等熵的instant刹时的streamline流线stream tube流管tangential切线的incompressible不行压缩的resultant合成的,组合的downstream下游的,顺水的elbow弯管,肘similitude相像性hydraulic水力的,水力学的predominante占主导地位spillway (河或水坝的)放水道,泄洪道prototype原型,样板Lesson 24 Mechanical Vibrationrepetitive重复的,频频的periodic周期的,按期的tidal潮的,像潮的stationary固定的,不动的vibratory振动的,摇动的propagation流传couplev .连结,连合acoustic听觉的,声学的annoyance烦忧,疑惑adjacent靠近的,周边的damp阻尼,衰减restore复职,送还neutral均衡exciting force激励力resonant adj.共振的,谐振的stiffness刚度,刚性proportionality成比率地inclusion包含,包含magnitude数值,大小substantially adv.本质上的数学英语词汇perturb搅乱,搅乱resonance n.共振vibratory adj.振动的,可知的perceptible可见的,可知的adudible听得见的,可闻的foregoing前述的impulsive冲击的shock冲击数学英语词汇Fourier series傅里叶级数excitation激发,激励discrete分别,失散的contend with向⋯作斗争compressor压气机fatigue疲惫perceptible可见的,可知觉的shredder切菜器disposal办理urban都市的metropolitan多数市的at-grade在同一水平面上elevated高架的guideway导轨Lesson 25 A prefect to the Continuum Mechanicspreface前言continuum连续pl. continuuarigid body刚体contemporary今世的,同期间的widespread散布广的,普及的accommodate容纳,使适应medium介质plasticity塑性residual节余的,残留的creep蠕变,爬行,塑性变形aging老化polymeric聚合(物)的sandy沙的,沙质的aubterranean地下的,隐蔽的essence精华,本质thermodynamics热力学self-similar自相像expedient方便的sonsolidate把⋯结合为一体,一致justify证明⋯有理数学英语词汇radically根当地,本质上deliberate冷静自在的,深谋远虑Lesson 33 what is a computerAttribute v.给予medieval中世纪的astronomer天文学家Mars火星数学英语词汇resemble vt.像,相像tedious adj.冗长无聊的pulp浆状物,果肉filter vt.过滤underlying adj.潜伏的,基本的ore n.矿沙,矿石perceive v.觉察,看见intervention n.干预,插入intelligent adj.有智力的,聪慧的Lesson 34 A computer systemmanipulate vt.操控,使用chip n.芯片etch vt.蚀刻,蚀镂fingernail指甲mount vt.安装,部署assemble vt.会合,齐集cabinet橱柜execute vt.履行,实现paycheck n.支付薪金的支票bar chart直方图joystick游戏杆encounter vt.碰到,碰上Mathematical Modelingindustryn.工业,家产,行业,勤劳commercen.商业complexityn.复杂(性),复杂的事物,复杂性careern. (原意:道路,轨道)事业,生涯,速度outsetn.初步,开始essencen.基本, [哲]本质,香精advocationn. (=advocacy)拥戴支持provisionn.供应, (一批)供应品,预备,防范,规定publicizev.宣扬roundaboutadj.迂回的,转弯抹角的n.道路交错处的环形路,迂回路线,兜圈子的话trial-errorvt. n.试制,试生产数学英语词汇maneuverabilityn.可操作性,灵活性vehiclen.交通工具,车辆,媒介物,传达手段junctionn.连结,接合,交错点,会合处ponderv.深思,考虑contrivev.发明,设计,图谋snookern. (=snooker pool)彩色台球,桌球contextn.上下文,文章的前后关系deviationn.背叛数学英语词汇notoriouslyadj.声名狼籍的。

A New Algebraization of the Lame Equation

uction
ψ ′′ (x) + E − m ℓ(ℓ + 1) sn2 x ψ (x) = 0 , (1.1)
The Lam´ e equation,
where ℓ is a real parameter1 , and sn x ≡ sn(x|m) is the usual Jacobian elliptic function of modulus m, occupies a central position in the theory of diﬀerential equations with periodic coeﬃcients. The study of its properties has attracted the attention of many illustrious mathematicians over the last century; classical references are [1–4]. Basic properties of the Lam´ e equation are as follows. First, it arises by separation of variables in the Laplace equation in ellipsoidal coordinates. Secondly, it possesses two linearly independent 2K (k ) or 4K (k )-periodic solutions (for characteristic √ values of E ) if and only if ℓ is a nonnegative integer. Here k = m, and K (k ) (denoted by K from now on) is the complete elliptic integral of the ﬁrst kind with parameter k :

数学难题

数学难题三等分角问题三等分角问题（trisection of an angle）是二千四百年前，古希腊人提出的几何三大作图问题之一，即用圆规与直尺把一任意角三等分。

问题的难处在于作图使用工具的限制。

古希腊人要求几何作图只许使用直尺（没有刻度，只能作直线的尺）和圆规。

这问题曾吸引着许多人去研究，但都无一成功。

１８３７年凡齐尔（１８１４－１８４８）运用代数方法证明了，这是一个标尺作图的不可能问题。

在研究”三等分角”的过程中发现了如蚌线、心脏线、圆锥曲线等特殊曲线。

人们还发现，只要放弃”标尺作图”的戒律，三等分角并不是一个很难的问题。

古希腊数学家阿基米得（前２８７－前２１２）发现只要在直尺上固定一点，问题就可解决了。

现简介其法如下：在直尺边缘上添加一点Ｐ，命尺端为Ｏ。

设所要三等分的角是∠ＡＣＢ，以Ｃ为圆心，ＯＰ为半径作半圆交角边于Ａ，Ｂ；使Ｏ点在ＣＡ延在线移动，Ｐ点在圆周上移动，当尺通过Ｂ时，联ＯＰＢ（见图）。

由于ＯＰ＝ＰＣ＝ＣＢ，所以∠ＣＯＢ＝∠ＡＣＢ／３。

这里使用的工具已不限于标尺，而且作图方法也与公设不合。

倍立方体问题倍立方体问题（problem of duplication of a cube ）是二千四百年前古希腊人提出的几何三大作图问题之一。

问题是指求作一立方体使其体积等于已知立方体体积的两倍。

本题难解的原因在于作图工具上有所限制，古希腊人强调几何作图只能用直尺（没有刻度，只能作直线的尺）和圆规。

关于倍立方问题的起源，有两个神话传说。

第一个是属于古希腊著名数学家、天文学家、哲学家埃拉托塞尼（前２７６－前１９５）的。

传说由于古希腊提洛岛（Delos ，爱琴海上小岛）上瘟疫流行，人们向太阳神第力亚祈祷，据说神要求把它殿前的祭坛的体积扩大一倍，而保持祭台的立方体形状不变。

因此，后人往往称倍立方体问题为提洛问题（Delos' problem）。

由于提洛岛上的居民并没有完成太阳神的”要求”，所以瘟疫也没有消除。

复化梯形算法求解数值积分

复化梯形算法求解数值积分摘要求某函数的定积分时，在多数情况下，被积函数的原函数很难用初等函数表达出来，因此能够借助微积分学的牛顿-莱布尼兹公式计算定积分的机会是不多的。

另外，许多实际问题中的被积函数往往是列表函数或其他形式的非连续函数，对这类函数的定积分，也不能用不定积分方法求解。

由于以上原因，数值积分的理论与方法一直是计算数学研究的基本课题。

构造数值积分公式最通常的方法是用积分区间上的n 次插值多项式代替被积函数，由此导出的求积公式称为插值型求积公式。

特别在节点分布等距的情形称为牛顿-柯茨公式，例如梯形公式与抛物线公式就是最基本的近似公式。

但是它们的精度较差。

而且高阶Newton-Cotes求积公式是不稳定的。

因此，通常不用高阶求积公式得到比较精确的积分值，而是将整个积分区间分段，在每一小段上用低阶求积公式。

这种方法称为复化求积方法。

本文从三个积分实例出发，主要讨论复化梯形公式以及精确程度分析。

关键词：数值积分；复化求积公式；复化梯形算法；MATLABTHE REHABILITATION OF TRAPEZOID FORMULA TO SOLVE THE NUMERICAL INTEGRATIONABSTRACTFind the definite integral of a function, in most cases, the original integrand function is difficult toexpress the elementary functions, it can use calculus of Newton - Leibniz formula to calculate thedefinite integral of the few opportunities . In addition, many practical problems in the integrand is often a list of functions or other forms of non-continuous function, the definite integral of suchfunctions, indefinite integral method can not solve. For these reasons, the numerical integration oftheory and method has been the subject of calculation of the basic mathematical research.Structural formula for numerical integration method is used most often on the n-th integration interval polynomial interpolation instead of the integrand, thus derived is called interpolation-typequadrature formula quadrature formula. Especially in the case of equidistant distribution of nodesis called Newton - Keci formula, such as trapezoidal formula and the formula is the most basicparabolic approximation formula. But their accuracy is poor. And high-level Newton-Cotesquadrature formula is unstable. So it is usually not higher-order quadrature formula to be moreprecise integral values, but the whole range of sub-points, with each short on low-level quadrature formula. This method is called complex method of quadrature.This example from three points of departure, the main complex of the trapezoid formula anddiscuss the accuracy of the analysis.Key words: Numerical integration；Rehabilitation of numerical integration；Rehabilitation of trapezoid formula；MA TLAB目录1 问题的提出 (1)2 问题的分析 (2)3 问题假设 (2)4 符号说明 (3)5 模型的建立及求解 (3)5.1 模型的准备工作 (3)5.1.1 复化梯形数值积分基本原理........... (3)5.2 模型的建立及求解 (4)6 模型验证及结果分析 (8)参考文献 (9)附录 (10)1问题提出有很多实际问题常常需要计算积分才能求解。

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Electronic Transactions on Numerical Analysis. Volume 16, pp. 143-164, 2003.Copyright©2003, Kent State University. ISSN 1068-9613.ETNAKent State University etna@A QUADRATURE FORMULA OF RATIONAL TYPE FOR INTEGRANDS WITHONE ENDPOINT SINGULARITYJ.ILL´ANAbstract.The paper deals with the construction of an efﬁcient quadrature formula of rational type to evaluate the integral of functions which are analytic in the interval of integration,except at the endpoints.Basically our approach consists in introducing a change of variable into the integralwhere and,,.We evaluate the new form by a quadrature approximant which is based on Hermite interpolation by means of rational functions.The nodes of are derived from a fundamental result proved by Ganelius[Anal.Math.,5(1979),pp.19-33]in connection with the problem of approximating the function,,by means of rational functions.Weﬁnd such that as,for all.For functions in,,which satisfy an integral Lipschitz condition of order,the following estimate is deducedIf then the upper bound for is that which is exact for the optimal quadrature error in,.We report some numerical examples to illustrate the behavior of the method for several values of the parameters.Key words.interpolatory quadrature formulas,rational approximation,order of convergence,boundary singu-larities.AMS subject classiﬁcations.41A25,41A55,65D30,65D32.1.Introduction.In applications the solution of a problem often involves the numeri-cal integration of functions with singularities.The traditional approach in this subject has been based on the use of Gauss quadrature formulas of polynomial type,though more re-cently some rational versions have also been considered.The latter concerns classes of Gauss quadrature formulas which require the largest degree of exactness for simple rational func-tions.These rational rules are connected with multipoint Pad´e approximation of Stieltjes functions[12,13,14,15,16,17,28],and with the evaluation of integrals whose integrand has poles close to the integration interval[7,8,9,10].Another relevant area is that concerned with the use of functions which transform the integration interval,and increase the efﬁciency of the numerical procedures.These trans-formations modify the distribution of the quadrature nodes in such a way that the new ones,namely,exhibit a higher concentration near those endpoints where the sin-gularities should be located.As far as we know the strategy ofﬁtting a change of variable into an integration rule to increase the efﬁciency of a numerical procedure,have only been investigated in the polynomial context[1,20,21,22,23,24,25].When compared to polynomials,rational functions are now considered a nicer class to approximate functions with a variety of singularities.This conclusion has a starting point in1964when Newman[26]proved that the function,,can be uniformly approximated by rational functions much faster than by polynomials.At present,there are additional reasons to assert that the following formulation(1.1)144A quadrature formula of rational type for integrands with one endpoint singularitywhere is a convenient rational function,is expedient when is an analytic function with a ﬁnite number of singularities on,or with some poles close to.A basic argument in favour of the latter is that the term in the error,can be better approximated than by polynomials,if annihilates the poles of closer to.The class of functions to be integrated are those deﬁned on the interval,,which admit analytic continuation to the space.Rational approximation of functions in a Hardy space was earlier investigated by Gonchar in[11]as a continuation to Newman’s research[26].In this paper we deal with a method of numerical integration which works,theoretically speaking,on the Hardy space and is based on interpolation by means of rational functions. For each this formula has the form given below(1.2)where,,,,,, ;is the-th derivative of,and is the quadrature error.Formula(1.2)is a special case of those studied by Bojanov[4],Newman[27],Andersson [2],Andersson&Bojanov[3].To understand why rational functions play a role in this theory it is sufﬁcient to consider the general formulation given below(1.3)where,is aﬁnite and positive measure on,is some ra-tional function of order with poles,,,and.It is well-known that is a continuous linear functional on,and that Cauchy’s integral formula can be applied to yield theﬁnite sum on the right term of(1.3).The exact bound,for functions in which satisfy an integral Lipschitz condition. Besides,we expect that the numerical behavior of such a procedure compares favorably with some other remarkable quadrature rules(cf.[20,23]).The paper is organized as follows.Section2is devoted to deﬁne nodes and coefﬁcients for integration formulas of the form(1.2).Section3deals with the construction of a change of variable which weﬁt into the integral,to reach an order of convergence which is optimal in according to the theoretical results obtained by Andersson[2].Finally, Section4contains some remarks and numerical examples to show the power of the method.2.Construction of the quadrature formula.The next result can be derived from[6] and is basic for our approach.J.Ill´a n145 L EMMA2.1.For,positive integer there exists a constant which only depends on,such that(2.1)where,;and for.Lemma2.1is based on the strong connection between rational approximation and some equilibrium problems.This result was used by Ganelius to obtain the exact order of conver-gence for the best uniform approximation of,,by means of rational functions of orden.Next we describe some of the most relevant aspects of Ganelius’technique to obtain estimate(2.1).Given the Green function of the right half plane,namelymust be assumed,and for allThe following step is to take as that given in Lemma2.1,toﬁnd(2.2)Notice that implies.At this point the nature of the number and the role played by it in this theory should be clear enough to the reader.Of course,much more details can be seen in the paper of Ganelius.After obtaining(2.2)the proof of lemma2.1is practically concluded.To derive the inequality(2.1)from(2.2),it only remains to replace the distribution by a discrete measure .Despite all the technical difﬁculties of this proof,the suitable location of the mass-points ,where the discrete distribution is supported,doesn’t seem very hard to obtain.The importance of the points for our work,comes from the fact that the nodes in(1.2) will be expressed in terms of the former.Andersson[2]applied Lemma2.1toﬁnd the exact rate of convergence of the optimal quadrature error in Hardy spaces.We also applied this result in Section3to obtain an upper146A quadrature formula of rational type for integrands with one endpoint singularitybound for the quadrature error generated by formula(1.2)after introducing a change of variable(see also[18]).The notation in Lemma2.1will be kept in the rest of the article.The proof of the following result is straightforward.L EMMA2.2.Let,,,and be real numbers such thatLet and be the circle with center at and radius .In addition,let consider the following rational functions(2.3)where is given bywith,,.Then,and satisfy the following properties1..2..3.is an increasing function on.4.,.5.If then.L EMMA2.3.Let and be the rational functions deﬁned in Lemma2.2.Let be the rational function given by the following equation(2.4)where is an analytic function in the unit disk.If is a non-negative and continuous function on then the rational function interpolates to at the zeros of,and the quadrature formula(2.5)is of the form(1.3)and it integrates exactly any rational function,J.Ill´a n147 where is the following polynomial of degree at mostLikewise,the zeros of are the poles of.They appear in the statement of the exactness condition for formula(2.5),and their formulation can be deduced from the following representation of(2.8)whose corresponding integrals can be evaluated by means of residues to obtain that all of them are zero.The inequality makes possible that for some values of and.It is the reason for which formula(2.8)can be considered as a suitable manner of representing the poles of.In case of it means that quadrature formula(2.5)is exact for some polynomials as well.We observe that quadrature formula(2.5)depends on the parameters,,,and. Moreover,if we put for some suitable,then we obtain a quadrature approximant of the same form as that in(1.2)with.3.An upper bound for the quadrature error.In this section we construct a special analytic change of variable,,to be introduced into the integral148A quadrature formula of rational type for integrands with one endpoint singularity Next we evaluate,with,, and,using the corresponding quadrature approximant (2.5)to approximate with an error of orderFor the function is a conformal mapping from the unit disk onto itself which satisﬁes and.Here we will only consider the case.To construct we need the following sequences.D EFINITION3.1.For every and,,,we deﬁne the sequence(3.1)J.Ill´a n1493.,,4.5., where and.Proof.It is trivial that property1is true for.For we use thatis a non decreasing function with respect to as well as the induction hypothesis on. The existence of the limit in property3is guaranteed by property2,and its value is obtained by taking limits on both sides of equation(3.1).After assuming that property5is true for we multiply both sides by the factor to obtain the relation for.Property6follows from the following equality(see(3.1))(3.4)(3.7)(3.9)150A quadrature formula of rational type for integrands with one endpoint singularityfrom which we obtain the following formula(3.10)Proof.Equation(3.12)is proved by induction on using Lemma3.2and the deﬁnition of both sequences and.Equality(3.12)can be seen as a consequence of applying times the change of variable (3.11)to the integral in the left hand side of(3.12).The approximation formula which is given in(3.13)is the result of coupling the approx-imant in(2.5)by putting,,and with the change of variable in(3.12).Theﬁnal version of our quadrature formula is(3.13)where(3.14)(3.15)J.Ill´a n151 where is the quadrature error whose expression is given in(2.9),and(3.17)Let be the rational function which produces the quadrature approxima-tion in(3.13).It means that is the rational function associated with according to the procedure given by(2.5).The weight function given by(3.17)corresponds to the statement of Lemma2.3,and will play a relevant role in applying Lemma2.1to our approach.Though we are mainly interested in evaluating the integral,,,we need the following result.L EMMA3.7.Let be a positive integer.Let,and.Letbe the quadrature approximant of formula(3.13)with nodes(2.7) for which the points are selected as those given by Lemma2.1.For sufﬁciently large, ,,,and the following estimate holds.(3.18)where is the circle with center at and radius.From property5of Lemma2.2,and(3.19)we obtain the following inequality.(3.20)152F IG.3.1.Error curves when,,,,.IGIn(3.20)we can consider the distributionJ.Ill´a n153 From Lemma(3.2)the distance from the circle to is. We apply to the principle of maximum to obtain(see[5],pp.29,36)(3.21)ing lemmas2.1and3.3,and equations(3.20),(3.21)and(3.22)we derive the following estimate.(3.23)where does not depend1on the parameters and.Let and where will be selected conveniently. From property4of Lemma2.2and Lemma3.4we obtainwhere.The terms in the right side of(3.24)have the following asymptotical behaviour as, respectively.1But it does depend on as can be deduced from[6].154A quadrature formula of rational type for integrands with one endpoint singularityF IG.3.3.Error surface when,,,,.The contribution of the factors,and leads us to select ,,in order to cancel all those terms which tend to inﬁnity as,with the exception of which is grouped together with.J.Ill´a n155T ABLE3.1Absolute errors when,,,.,0.0000000e+00 2.0e–010.0000000000e+009.8e–031.0000000e–012.0e–01 2.0000000000e–018.4e–033.0000000e–01 1.8e–01 5.0000000000e–01 6.1e–035.0000000e–01 1.6e–017.0000000000e–01 4.2e–037.0000000e–01 1.2e–018.0000000000e–01 3.0e–037.5000000e–01 1.1e–019.0000000000e–01 1.5e–038.0000000e–018.2e–029.5000000000e–01 3.4e–048.9000000e–019.5e–039.5910000000e–01 2.5e–068.9700000e–01 4.4e–049.5915000000e–01 4.1e–078.9730000e–01 3.5e–059.5915900000e–01 3.8e–088.9732485e–01 1.7e–069.5915970654e–01 2.3e–100.000000000000000e+00 6.4e–072.000000000000000e–01 1.2e–019.999000000000000e–01 5.9e–079.900000000000000e–01 3.4e–039.999300000000000e–01 5.6e–079.997260000000000e–01 4.6e–079.999600000000000e–01 5.1e–079.997265635000000e–01 1.2e–109.999942747252746e–01 1.2e–119.997265635029354e–01 3.5e–12D EFINITION3.8.The integral modulus of continuity of is given by(3.25)The analysis of error in terms of the modulus(3.25)is the following(3.26)It is well-known that,for all,and(cf.[5,18]). Besides,the behaviour of as does not depend essentially on,, but on the nature of in a neighbourhood of the endpoint.T HEOREM3.9.Let,,,and, .Let be a sequence such that1.,2.,3.156A quadrature formula of rational type for integrands with one endpoint singularityT ABLE3.3Absolute errors when,,,,.0.000000000000000e+00 2.1e–092.500000000000000e–018.5e–103.000000000000000e–01 1.9e–097.500000000000000e–01 3.5e–107.000000000000000e–01 1.6e–099.500000000000000e–019.8e–119.900000000000000e–01 1.4e–099.960000000000000e–01 5.5e–129.999571142857140e–017.5e–129.989981684981686e–01 2.0e–15is the approximant given by(3.13),and the positive constant depends neither on nor.Convergence follows from estimate(3.27).J.Ill´a n157F IG.3.7.Error for,,,,,,.If then every sequence of the form,satisﬁes the conditions of Theorem3.9.Besides,the rate of convergence of the quadrature error with respect to depends on the behavior of the sequenceThe following result states a class of functions for which we should expect a good behavior158A quadrature formula of rational type for integrands with one endpoint singularityT ABLE3.4Absolute errors when,,,,.0.000000000000e+00 3.2e–020.0000000e+00 5.8e–055.000000000000e–01 3.0e–02 2.0000000e+00 4.9e–057.000000000000e–01 1.4e–02 5.0000000e–01 2.1e–057.500000000000e–01 5.7e–03 6.0000000e–01 2.8e–067.710000000000e–01 1.4e–03 6.1000000e–01 5.1e–077.750000000000e–01 5.4e–04 6.1210000e–01 1.3e–087.760000000000e–01 3.1e–04 6.1213000e–01 6.3e–097.773000000000e–01 1.1e–05 6.1213307e–01 5.6e–097.773300000000e–01 4.0e–067.0000000e–01 2.7e–057.773385518591e–01 2.0e–069.0000000e–01 2.5e–04in the proof of Lemma3.7.We observe that we must consider for,and otherwise.Now the term in estimate (3.27)becomesLet be a function in such that it satisﬁes a Lipschitz condition of order.Ifand,,then we haveJ.Ill´a n159T ABLE3.5Absolute errors when,,,,,.0.000000000000e+00 6.6e–061.00000000e–012.2e–055.000000000000e–01 2.9e–064.00000000e–01 1.3e–059.000000000000e–01 4.1e–078.00000000e–01 2.9e–069.950000000000e–017.4e–099.15200000e–01 1.8e–089.970800000000e–01 2.9e–119.15823525e–01 2.8e–10T ABLE3.6Absolute errors when,,,,.0.000000000000000e+00 6.6e–062.00000000000000e–01 5.0e–069.500000000000000e–01 1.8e–079.00000000000000e–01 4.1e–079.995000000000000e–01 1.7e–099.99967920000000e–01 1.3e–109.999787058820000e–01 6.1e–129.99967921568627e–01 2.1e–130.000000000e+008.4e+001.000000000e–018.3e+003.000000000e–018.1e+005.000000000e–017.8e+007.000000000e–017.5e+009.000000000e–017.1e+009.999990000e–01 2.3e+009.999999500e–018.3e–019.999999900e–01 2.6e–029.999999905e–01 4.8e–04T ABLE3.8Absolute errors when,,,,,.0.00000000000000e+008.7e+001.00000000000e–018.6e+002.00000000000000e–018.5e+004.00000000000e–018.3e+006.00000000000000e–018.0e+008.00000000000e–017.5e+009.99000000000000e–01 4.6e+009.99999999000e–018.0e–019.99999999970000e–018.3e–019.99999999919e–01 6.9e–03160A quadrature formula of rational type for integrands with one endpoint singularityT ABLE3.9Absolute errors when,,,,.0.00000000000000e+008.7e+001.0000000000000e–018.4e+005.00000000000000e–018.1e+005.0000000000000e–018.1e+009.00000000000000e–017.0e+009.0000000000000e–017.0e+009.99999000000000e–01 2.5e+009.9999999000000e–01 1.6e+009.99999999958000e–019.2e–019.9999999994258e–01 1.6e–010.000000e+00 2.7e–011.000000e–012.5e–012.000000e–01 2.2e–013.000000e–01 2.0e–014.000000e–01 1.7e–016.000000e–019.6e–028.000000e–01 2.1e–038.039000e–01 4.1e–058.039760e–01 3.3e–078.039765e–01 6.6e–08T ABLE3.11Absolute errors when,,,,,.0.00000e+00 2.4e–011.0000000e–012.2e–012.00000e–01 2.0e–016.0000000e–01 1.2e–016.00000e–01 1.2e–019.9000000e–01 5.1e–039.90000e–01 5.2e–039.9998100e–01 2.4e–059.99990e–01 1.0e–059.9998158e–01 2.6e–07T ABLE3.12Absolute errors when,,,,,.0.000000000000e+00 6.4e–111.0e–01 1.4e–102.000000000000e–01 4.8e–113.0e–01 1.3e–106.000000000000e–01 2.7e–115.0e–01 1.2e–108.000000000000e–01 1.7e–117.0e–01 1.0e–109.780000000000e–01 1.1e–129.0e–019.2e–11J.Ill´a n161T ABLE3.13Absolute errors when,,,.,0.00000000e+00 1.4e–010.0000e+00 3.5e–021.00000000e–01 1.4e–01 1.0000e–012.9e–023.00000000e–01 1.3e–01 2.0000e–01 2.3e–024.00000000e–01 6.0e–02 3.0000e–01 1.8e–025.00000000e–01 1.3e–01 5.0000e–019.9e–039.00000000e–017.8e–027.0000e–01 3.8e–039.72000000e–01 1.8e–039.0000e–01 1.2e–039.72700000e–01 6.0e–069.3000e–01 5.0e–049.72702200e–01 2.3e–079.3950e–01 2.6e–059.72702285e–01 2.0e–099.3991e–01 5.2e–080.000000000e+00 1.9e–091.000000e–012.9e–022.000000000e–01 1.2e–095.000000e–01 1.0e–025.000000000e–01 3.8e–109.000000e–01 5.5e–047.000000000e–01 3.7e–119.952700e–017.6e–097.318000000e–01 4.8e–149.952725e–019.8e–14we only refer to the case.By means of the afﬁne transformation,,we apply our method to estimate, for several values of and.If then it means that no change of variable is made.The values of the parameter which we have used to test our approach are. In spite of the theoretical results in the previous section we have obtained very good results for,though instability shows up for in case of.In this article we report Fig.3.4for,Fig.3.7to validate the role of in the procedure and Table3.13for the case.We use the exactness condition for the rational functions to implement a numerical procedure for the quadrature rule(3.13).It means that the coefﬁcientsof the approximant are calculated as the solution of a lin-ear system of equations which we trasform in,where is an upper triangular matrix obtained via QR factorization.Table3.17shows that after scaling the con-dition number of should vary a little as ranges from to.Many numerical experiments have shown that the equation works good enough in the computer.In our opinion this procedure needs a parameter which ranges from to.If we make small it means that either the value of is very forced to be near the point,which yields a very high concentration of nodes and poles near the point,or.Instability is associated with small values of though it can be observed higher accuracy as well(see Fig.3.5-3.6).Theorem3.10shows that if is close enough to one then the quadrature error should be small.This effect is certainly produced by the change of variable,though one can detect that the decreasing behaviour of the162A quadrature formula of rational type for integrands with one endpoint singularityT ABLE3.15Absolute errors when,,,,.0.00000000000000e+00 1.6e–081.00e–01 3.8e–062.00000000000000e–01 1.1e–084.00e–01 1.8e–064.00000000000000e–01 6.1e–097.00e–01 4.8e–077.00000000000000e–01 1.6e–099.00e–01 6.1e–089.90000000000000e–01 2.2e–129.99e–01 1.3e–130.0000e+00 6.4e–031.0000000e–017.5e–022.0000e–01 5.5e–035.0000000e–01 5.5e–025.0000e–01 3.9e–038.0000000e–01 3.2e–028.0000e–01 2.0e–039.9000000e–01 2.0e–039.9700e–01 3.8e–059.9234833e–01 2.3e–06error with respect to is magniﬁed a lot by the condition.When is too small the slope of the error curve with respect to is also small except for values of very close to one.Such a behaviour can be seen in Table3.1–3.16for several values of the parameters, ,and.The variable plays the role of counterpart of,in the sense that values of close to produce a concentration of nodes on the right side.We displayﬁgures3.1–3.6to illustrate the behavior of the error as function of the parameters and,particularly when they are close to one simultaneously.The error surfaces have been deﬁned over the rectangular grid,,,for which,.Before making any conclusion on whether the selection is a reasonable decision for the numerical procedure one has to take into account that condition implies that some derivatives of the integrand are participating in the calculations.Despite the fact of having used simbolic tools to simulate all the derivatives,it is not surprising that a loss of accuracy is observed in Table3.16with respect to Table3.2.Naturally,this comparison clearly indicates that unless we were able to improve the algorithm,the case seems to be preferable.As for the use of an integral representation formula to calculate all the coefﬁcients of the quadrature rule(3.13),for the moment we have not been able to reach precision enough in the experiments.For all cases a feature of the integration method(3.13) is that the error strongly depends on the behavior of near.A class of functions with singularities located at interior points to which can be applied this integration rule,is one as that deﬁned in[19],namely,piecewise functions.J.Ill´a n163T ABLE3.17Condition number of matrices,.1.2e+00 4.1e+00 4.9e+012.6e+03 1.9e+067.7e+10 1.4e+130.11.6e+007.0e+008.4e+01 5.2e+03 4.3e+062.0e+117.7e+130.91.7e+019.1e+018.8e+02 1.9e+05 4.6e+08 5.0e+13 1.3e+170.011.2e+00 1.8e+00 4.5e+01 6.9e+03 4.2e+07 6.3e+12 6.6e+190.52.0e+00 4.2e+033.8e+049.0e+107.7e+12 1.7e+19 3.7e+230.99The reader should consult[23]where the-point Gauss-Legendre rule and several smooth-ing transformations are applied together to evaluate the integral of functions with one and two endpoint singularities.To illustrate the efﬁciency of the method in[23],the authors compare the corresponding numerical results with those generated by the trapezoidal rule when the latter has been modiﬁed by a change of variable.Our approach is based on interpolation of rational functions and it is different from that presented by Monegato and Scuderi in[23].The latter deals with Gaussian quadrature for-mulas of polynomial type and smoothing transformations for which some derivatives van-ish at the endpoints of the integration interval.Instead,we consider the change of variable which simply modiﬁes the distribution of nodes to diminish the adverse effect of the endpoint singularity.However,for the purposes of comparison,we mainly refer to the numerical results in[23]because here we have practically tested the same functions as those reported in that paper.In spite of the ill conditioned matrices which arise in our implementation,a conclusion is that the accuracy which we can obtain with the rational quadrature rule(3.13)is competent, particularly for those functions for which estimate(3.28)holds.All the computations in this work have been performed on a PC using MatLab. Acknowledgments.The author would like to thank the referees,who spotted many minor errors and offered valuable suggestions.REFERENCES[1] A.A IMI,M.D ILIGENT,AND G.M ONEGATO,New numerical integration schemes for applications ofGalerkin BEM to2-D problems,Internat.J.Numer.Methods Engrg.,40(1997),pp.1977–1999.[2]J.E.A NDERSSON,Optimal quadrature of functions,Math.Z.,172(1980),pp.55–62.[3]J.E.A NDERSSON,AND B.D.B OJANOV,A Note on the optimal quadrature in,Numer.Math.,44(1984),pp.301–308.[4] B.D.B OJANOV,On an optimal quadrature formula,C.R.Acad.Bulgare Sci.,27(1974),no.5,pp.619–621.[5]P.L.D UREN,Theory of Spaces,Academic Press,New York,p.157,1970.[6]T.G ANELIUS,Rational approximation to on,Anal.Math.,5(1979),pp.19–33.[7]W.G AUTSCHI,Gauss-type quadrature rules for rational functions,in“Numerical Integration IV”(H.Brassand G.H¨a mmerlin,Eds.),Internat.Series of Numerical Mathematics,112,Birkh¨a user,Basel(1993),pp.111–130.[8],The use of rational functions in numerical quadrature,put.Appl.Math.,133(2001),pp.111–126.[10]W.G AUTSCHI,L.G ORI,AND M.L.L O C ASCIO,Quadrature Rules for Rational Functions,Numer.Math.,86(2000),pp.617–633.164A quadrature formula of rational type for integrands with one endpoint singularity[11] A.A.G ONCHAR,On the rate of rational approximation to continuous functions with characteristic singulari-ties,Mat Sb.,73(115)(1967),pp.630–638;[SR Sb.,2(1967),pp.561–568.][12] A.A.G ONCHAR,AND G.L´OPEZ-L AGOMASINO,On Markov’s theorem for multipoint Pad´e approximants,Mat.Sb.,105(147)(1978),pp.512–524;[SR Sb.,34(1978),pp.449–459.][13]P.G ONZ´ALEZ-V ERA,M.J IMENEZ P AIZ,R.O RIVE,AND G.L´OPEZ-L AGOMASINO,On the convergenceof quadrature formulas connected with multipoint Pad´e–type approximation,J.Math.Anal.Appl.,202(1996),pp.747–775.[14] F.C ALA R ODR´I GUEZ,P.G ONZ´ALEZ-V ERA,M.J IM´E NEZ P AIZ,Quadrature formulas for rational func-tions,Electron.Trans.Numer.Anal.,9(1999),pp.39–52./vol.9.1999/pp39-52.dir/pp39-52.pdf.[15]J.I LL´AN,AND G.L´OPEZ-L AGOMASINO,Quadrature formulas for unbounded intervals,Cienc.Mat.(Ha-vana),3(1982),no.3,pp.29–47(in Spanish).[16],Numerical integration based on interpolation and their connection with rational approximation,Cienc.Mat.(Havana),8(1987),no.2,pp.31–44(in Spanish).[18]J.I LL´AN,On the rational approximation of functions in the metric,in Approximation and Op-timization,A.G´o mez,F.Guerra,M.A.Jim´e nez and G.L´o pez,Eds.,Lecture Notes in Math.,1354,Springer Verlag,1987,pp.155–163.[19],High order methods for weakly singular integral equations with non smooth input functions,Math.Comput.,67(1998),pp.1493–1515.[25]M ASATAKE M ORI,AND M ASAAKI S UGIHARA,The double–exponential transformation in numerical anal-ysis,put.Appl.Math.,127(2001),pp.287–296.[26] D.J.N EWMAN,Rational approximation to,Michigan Math.J.,11MR30#1344(1964),pp.11–14.[27]。