7097044-Fractional-Calculus
fractal and fractional佩普学术 -回复
fractal and fractional佩普学术-回复Fractal and Fractional PEP AcademicIntroduction:Fractals and fractions are two mathematical concepts that have significant applications in various fields, including physics, computer graphics, and finance. This article aims to provide a comprehensive understanding of fractals and fractions, exploring their basic definitions, properties, and real-life applications.I. Fractals:1. Definition:Fractals are geometric shapes that exhibit self-similarity, meaning that they contain smaller copies of themselves in a never-ending pattern. They can be generated through a mathematical process called recursion. Examples of well-known fractals include the Mandelbrot set and the Sierpinski triangle.2. Properties:Fractals possess several distinctive properties, including infinite complexity, fractional dimension, and non-integer scaling. These properties contribute to their unique visual appearance and make them applicable in various fields, such as computer graphics and image compression.3. Applications:Fractals find applications in many practical areas. In computer graphics, they are used for creating realistic landscapes, textures, and natural objects. Fractal-based algorithms are also employed in image compression techniques, enabling efficient storage and transmission of digital images. Additionally, fractal analysis is utilized in medical imaging, financial forecasting, and weather prediction.II. Fractions:1. Definition:Fractions are numerical expressions representing a part or parts ofa whole. They consist of a numerator and a denominator, with the numerator representing the number of parts involved and the denominator indicating the total number of equal parts that make up the whole. For example, 3/4 represents three parts out of four equal parts.2. Properties:Fractions possess various properties, including equivalence, addition, subtraction, multiplication, and division. Equivalent fractions represent the same part-to-whole ratio, while adding, subtracting, multiplying, or dividing fractions follow specific rules and algorithms.3. Applications:Fractions have numerous real-life applications. In cooking and baking, fractions are used to determine ingredient quantities accurately. In finances, fractions are utilized to calculate interest rates, percentages, and financial ratios. Moreover, fractions play a significant role in measurements, allowing precise representations of lengths, weights, and volumes.III. Fractals and Fractions:1. Fractional Crystals:Fractional crystals are a special type of fractal pattern that combines the concepts of fractals and fractions. They are formed by repeatedly replacing parts of a shape with smaller copies. Each iteration involves dividing the shape into fractions of the original size and replacing them with smaller-scale copies.2. Applications:Fractional crystals offer an effective way to represent complex structures with fractional dimensions. They find applications in physics, chemistry, and materials science. For instance, they are used to model the behavior of polymers, the structure of porous materials, and the properties of amorphous solids.Conclusion:Fractals and fractions are fundamental mathematical concepts withsignificant practical applications. Fractals exhibit self-similarity and possess unique properties, making them useful in computer graphics, image compression, and numerous scientific fields. Fractions, on the other hand, represent parts of a whole and find applications in cooking, finance, and measurements. The combination of fractals and fractions leads to the concept of fractional crystals, enabling the representation of complex structures with fractional dimensions. Understanding these concepts is essential for anyone interested in mathematics or its various applications.。
The Fundamental Theorem of Calculus
P (x)dx
= (165x − 0.05x2 )|1600 1500 = 1000 . Thus, increasing monthly production from 1,500 units to 1,600 units will increase the monthly profit by $1,000. (See College Mathematics for Business, §12-5, Example 5.)
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§6-2 The Fundamental Theorem of Calculus
Applications of Definite Integrals
Example (Useful life)
An amusement company maintains records for each video game it installs in an arcade. Suppose that C(t) and R(t) represent the total accumulated costs and revenues (in thousands of dollars), respectively, t years after a particular game has been installed and that C (t) = 2 , R (t) = 9e−0.5t .
Example (Useful life)
An amusement company maintains records for each video game it installs in an arcade. Suppose that C(t) and R(t) represent the total accumulated costs and revenues (in thousands of dollars), respectively, t years after a particular game has been installed and that C (t) = 2 , R (t) = 9e−0.5t .
2024年AP Calculus AB真题集锦
2024年AP Calculus AB真题集锦在2024年的AP Calculus AB考试中，考生们将面临一系列挑战，需要全面掌握各个知识点和技巧来应对。
本文将为大家提供一个真题集锦，帮助大家更好地备考。
第一部分：选择题选择题是AP Calculus AB考试中的重要组成部分，考察学生对基础知识的理解和应用能力。
在这个部分，我们将精选几道典型的选择题，供大家练习。
题目1：设函数f(x)为连续函数，且在区间[0,1]上满足f(0)=0，f(1)=2。
已知f'(x)>0，且在[0,1]上有f''(x)<0，则下列哪一个选项是正确的？A. f(x)在(0,1)内取值为负数B. f(x)在(0,1)内取值为正数C. f(x)在(0,1)内取值为0D. 无法确定f(x)在(0,1)内的取值解析：根据题意可知，函数f(x)在[0,1]上是单调递增的，同时具有凹向下的图像形状。
因此，f(x)在(0,1)内取值为正数，选项B为正确答案。
题目2：已知函数f(x)=2x^3-3x^2-12x+5，求f'(2)的值。
A. 12B. 14C. 16D. 18解析：首先，我们对f(x)进行求导，得到f'(x)=6x^2-6x-12。
将x=2代入f'(x)中，得到f'(2)=6(2)^2-6(2)-12=12。
因此，选项A为正确答案。
第二部分：解答题解答题是AP Calculus AB考试中的较为复杂和综合性要求较高的部分，需要考生们综合运用所学知识来解决问题。
下面，我们选择两道典型的解答题供大家参考。
题目3：已知函数f(x)在区间[-1,3]上连续，且满足f'(x)=2(x-1)和f(1)=4。
若曲线y=f(x)与x轴所围成的面积为8，求函数f(x)在区间[-1,3]上的最小值和最大值。
解析：首先，根据题意可以确定f(x)的一阶导数为f'(x)=2(x-1)，再对f'(x)进行积分，得到f(x)=x^2-2x+C，其中C为常数。
AP Calculus AB考试2024历年题目全解
AP Calculus AB考试2024历年题目全解2024年的AP Calculus AB考试历年题目全解[注意：本文以解题步骤为主要内容，为了更好地呈现题目解析，会包含大量的公式和符号。
请确保在阅读时有一定的数学基础。
]1. 第一题题目描述：给定一个函数 f(x)，求其在区间 [a，b] 上的定积分。
解析：根据题目所给函数 f(x) 的表达式，我们可以使用不定积分的方法解决这个问题。
首先，先求 f(x) 的原函数 F(x)，然后计算 F(b) -F(a) 即可得到定积分的结果。
具体步骤如下：1. 求函数 f(x) 的原函数 F(x)。
2. 计算 F(b) - F(a) 的值，即为所求定积分的结果。
2. 第二题题目描述：计算函数 f(x) 的导数。
解析：根据题目所给函数 f(x) 的表达式，我们可以使用求导的方法来计算其导数。
根据导数的定义，导数 f'(x) 可以通过计算 f(x) 的极限来得到。
具体步骤如下：1. 对函数 f(x) 求导。
2. 化简导数的表达式，得到最终结果。
3. 第三题题目描述：给定一个函数 f(x)，求其在某点 x=c 处的极限。
解析：根据题目所给函数 f(x) 的表达式，我们可以使用极限的定义来计算其在某点 x=c 处的极限。
根据极限的定义，我们需要分别计算 x 无限接近 c 时 f(x) 的左极限和右极限。
具体步骤如下：1. 计算 f(x) 在 c 的左极限。
2. 计算 f(x) 在 c 的右极限。
3. 判断左极限和右极限是否存在且相等，若相等则极限存在，否则不存在。
4. 第四题题目描述：给定一个函数 f(x)，求其在某点 x=c 处的导数。
解析：根据题目所给函数 f(x) 的表达式，我们可以使用导数的定义来计算其在某点 x=c 处的导数。
导数 f'(x) 可以通过计算 f(x) 的极限来得到。
具体步骤如下：1. 对函数 f(x) 求导。
2. 将 x 的值替换为 c，计算导数的值，即为所求导数的结果。
2012国际Caculus(I)期中(上)考试卷-副本(2)
江西财经大学2012级期中（上）考试试卷课程代码：12003（A ） 授课课时：48 考试时间：110分钟 课程名称：Calculus (I) 适用对象：12级国际学院本科生 试卷命题人： 聂 高 辉 . 试卷审核人： 聂 高 辉 .1. Full in the blank of each statement in the following five statements such that the statement is right. Then write the corresponding answer on the answer book by the title number. You can be gained 3 points for the right thing on per blank.(1) (sec )x '=________.(2) 0sin 2limx x x→=_________. (3) d (log )d a x x =________. (4) 2lim 1x x x →∞⎛⎫+= ⎪⎝⎭________.(5) The tangent line equation of the cure ()f x =(1,1)is________.2. Choose the one that the statement is right from four choices marked A, B, C, and D, with which each statement of the following five statements. Then write the corresponding answer on the answer book by the title number. You can be gained 3 points for the right choice of per the statement.(6)1d |x z ==________provided that 2z x =(A) d x . (B) 2d x . (C) 0. (D) 2. (7) If (1)2f =, (1)6f '=and (2)8f '=, then 12d ()|d x f x x-== . (A) 1/8. (B) 1/6. (C) 1/2. (D) 1.(8) Differentiable functions ()f x and ()g x with ()()[,]f x g x x a b <∈hold property for which .(A) ()()[,]f x g x x a b ''<∈ (B) ()()[,]f x g x x a b ''≤∈(C) 000lim ()lim ()(,)x x x x f x g x x a b →→≤∈ (D)000lim ()lim ()(,)x x x x f x g x x a b →→<∈ (9) If sin lim x x a x →∞=and 01lim sin x x b x→=, then (,)a b =________ (A).(1,1). (B). (1,0). (C).(0,1). (D) (0,0).(10) If 0()2f x =, then in the following propositions is right.(A)0lim ()2x x f x →=. (B)0()0f x '=. (C)0lim ()2x x f x →≠. (D)0d (())0d f x x=. 3. Give the solution of any of in the following problems and write operation process and answer on the answer book by the title number. You can be gained mark of per problem by the right process and answer.(11) (10pts) Find 21x →and 2lim()x x →∞ (12) (6pts) 210lim(sec )x x x → (13) (6pts) Evaluate 2lim sin 1x x x x →∞+(14) (8pts). Find d d y x if sin sin x x y x x =(15) (6pts). Compute x + (16) (8pts) Find d d y x , 22d d x yif xy x y e += (17) (8pts) Find d d n n z xif (21)ln(21)z x x =++ (17). (12pts) A company estimates that the demand function of an item on the market is ()2003D p p =-.(1) Find marginal demand at 2p =and interpret its meaning? (2)If the formula ()()D p p D p 'is said to be elasticity of demand with respective to price at p , then what is the elasticity of demand of this item at 2p =? Can you say meaning of the elasticity?(3)Please you give the elasticity formula of a general function ()f x .(18) (6pts) Prove that )(x f is continuous at 0x if and only if 00lim[()()]0x x f x f x →-=。
AP Calculus AB历年真题精选2024
AP Calculus AB历年真题精选2024 2024年的AP Calculus AB考试即将到来，对于准备参加考试的学生来说，熟悉历年真题是非常重要的。
在本文中，我将为大家精选几道2024年AP Calculus AB考试的历年真题，帮助大家更好地备考。
一、选择题1. 求函数f(x) = x^3 - 3x^2 + 2x的导数。
解析：根据导数的定义，对函数f(x)求导数，我们可以将每一项的指数乘以系数，并减少指数。
因此，f'(x) = 3x^2 - 6x + 2。
2. 设函数g(x) = (x - 2)^2 + 3，求g(x)的最小值。
解析：由于(x - 2)^2的值永远大于等于0，所以g(x)的最小值就是当(x - 2)^2的值为0时，即x = 2。
代入函数g(x)，可以得到最小值为3。
二、计算题1. 求函数y = x^2 + 4x + 3的定积分，区间为[-1, 2]。
解析：我们可以先对函数y = x^2 + 4x + 3求不定积分，得到原函数F(x) = (1/3)x^3 + 2x^2 + 3x + C。
然后，我们将上下限带入原函数，得到积分值为F(2) - F(-1) = 11。
2. 求曲线y = x^3在x = 2处的切线方程。
解析：首先，我们需要求得曲线y = x^3的导数，即y' = 3x^2。
然后，代入x = 2，得到斜率k = 12。
由于切线方程的一般形式为y = kx +b，我们可以代入已知的点(x, y) = (2, 8)，解得b = -4。
因此，切线方程为y = 12x - 4。
三、证明题证明：对于任意实数x和y，有(x + y)^2 ≥ 4xy。
解析：我们可以展开(x + y)^2，得到x^2 + 2xy + y^2。
根据平方项的非负性，我们可以得出x^2 ≥ 0和y^2 ≥ 0。
因此，x^2 + y^2 ≥ 0，进而有x^2 + y^2 + 2xy ≥ 2xy。
AP_calculus参考书
Score
AP Score 5 4 3 2 1 Qualification Extremely well qualified Well qualified Qualified Possibly qualified No recommendation
Topic Outline for Calculus BC
1
by taking sufficiently close values of the domain.) • Understanding continuity in terms of limits. • Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem). * Parametric, polar, and vector functions. The analysis of planar curves includes those given in parametric form, polar form, and vector form. II. Derivatives Concept of the derivative • Derivative presented graphically, numerically, and analytically. • Derivative interpreted as an instantaneous rate of change. • Derivative defined as the limit of the difference quotient. • Relationship between differentiability and continuity. Derivative at a point • Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents. • Tangent line to a curve at a point and local linear approximation. • Instantaneous rate of change as the limit of average rate of change. • Approximate rate of change from graphs and tables of values. Derivative as a function • Corresponding characteristics of graphs of ƒ and ƒ∙. • Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ’. • The Mean Value Theorem and its geometric interpretation. • Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. Second derivatives • Corresponding characteristics of the graphs of ƒ, ƒ’, and ƒ’’∙. • Relationship between the concavity of ƒ and the sign of ƒ’∙. • Points of inflection as places where concavity changes. Applications of derivatives • Analysis of curves, including the notions of monotonicity and concavity. + Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration. • Optimization, both absolute (global) and relative (local) extrema. • Modeling rates of change, including related rates problems. • Use of implicit differentiation to find the derivative of an inverse function. • Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration. • Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations. + Numerical solution of differential equations using Euler’s method. + L’Hospital’s Rule, including its use in determining limits and convergence of improper integrals and series. Computation of derivatives • Knowledge of derivatives of basic functions, including power, exponential, logarithmic,
第二单元参考书目-中国科学院数学与系统科学研究院
现代微分几何
李洪波、刘卓军
《微分几何讲义》，陈省身，陈维桓著。
纠错码理论
万哲先、黄民强、邓映蒲、 《代数和编码》万哲先著，科学出版社。 F.J.MacWillams and N.J.A. Sloane, “The Theory of Error
刘卓军
–Correcting Codes (PartⅠ)” 。
第三单元参考书目
学科专业
第三单元
（二级学科）
（专业课）
代数表示论
李代数
代数拓扑
数论
代数几何
偏微分方程（乙）
动力系统 系统与控制理论
数学物理
基础数学 复动力系统与值分布
论
多复变与复几何
动力系统 偏微分方程（乙）
算子代数
偏微分方程
数论 常微分方程
导师
考试覆盖内容及参考书
韩阳
席南华、赵开明、徐晓平
段海豹、潘建中
高全泉
同运筹学与控制论 同系统建模与控制 《人工智能》(上、下册)，陆汝钤著，科学出版社。 Roger S. Pressman, Software Engineering A Practitioner’s Approach (软件工程：实践者的研究方法)， R. S. Pressman & Associates 公司，机械工业出版社。
学出版社, 2005), Chapters 11 & 19
杨乐、王跃飞、崔贵珍
周向宇
贺正需 张平
葛力明
丁彦恒 张志涛 葛力明 郑作环
F.Warner: Foundations of differential manifolds and Lie groups, GTM94。 Griffiths,Harris: Principles of algebraic geometry. John Wiley & Sons, Inc., 1978; Fritzsche, Grauert: From holomorphic functions to complex manifolds. GTM213, Springer, 2002。
calculus我的calculus笔记,按照国际标准排版
1.2
Mathematical Models: A Catalog of Essential Functions
1. A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon. 2. If there is no physical law or principle to help us formulate a model, we construct an empirical model. 3. A function P is called a polynomial if P pxq an xn an¡1 xn¡1 ¤ ¤ ¤ a2 x2 a1 x a0 when n N and the numbers a0 , a1 , a2 , . . . , an are constants called coefficients of the polynomial. The domain of any polynomial is R. If the leading coefficient an $ 0, then the degree of the polynomial is n. A polynomial of degree 1 is of the form P pxq mx b and is a linear function. A polynomial of degree 2 is of the form P pxq ax2 bx c and is called a quadratic function.
1
CHAPTER 1
斯普林格数学研究生教材丛书
《斯普林格数学研究生教材丛书》（Graduate Texts in Mathematics）GTM001《Introduction to Axiomatic Set Theory》Gaisi Takeuti, Wilson M.Zaring GTM002《Measure and Category》John C.Oxtoby（测度和范畴）（2ed.）GTM003《Topological Vector Spaces》H.H.Schaefer, M.P.Wolff（2ed.）GTM004《A Course in Homological Algebra》P.J.Hilton, U.Stammbach（2ed.）（同调代数教程）GTM005《Categories for the Working Mathematician》Saunders Mac Lane（2ed.）GTM006《Projective Planes》Daniel R.Hughes, Fred C.Piper（投射平面）GTM007《A Course in Arithmetic》Jean-Pierre Serre（数论教程）GTM008《Axiomatic set theory》Gaisi Takeuti, Wilson M.Zaring（2ed.）GTM009《Introduction to Lie Algebras and Representation Theory》James E.Humphreys（李代数和表示论导论）GTM010《A Course in Simple-Homotopy Theory》M.M CohenGTM011《Functions of One Complex VariableⅠ》John B.ConwayGTM012《Advanced Mathematical Analysis》Richard BealsGTM013《Rings and Categories of Modules》Frank W.Anderson, Kent R.Fuller（环和模的范畴）（2ed.）GTM014《Stable Mappings and Their Singularities》Martin Golubitsky, Victor Guillemin （稳定映射及其奇点）GTM015《Lectures in Functional Analysis and Operator Theory》Sterling K.Berberian GTM016《The Structure of Fields》David J.Winter（域结构）GTM017《Random Processes》Murray RosenblattGTM018《Measure Theory》Paul R.Halmos（测度论）GTM019《A Hilbert Space Problem Book》Paul R.Halmos（希尔伯特问题集）GTM020《Fibre Bundles》Dale Husemoller（纤维丛）GTM021《Linear Algebraic Groups》James E.Humphreys（线性代数群）GTM022《An Algebraic Introduction to Mathematical Logic》Donald W.Barnes, John M.MackGTM023《Linear Algebra》Werner H.Greub（线性代数）GTM024《Geometric Functional Analysis and Its Applications》Paul R.HolmesGTM025《Real and Abstract Analysis》Edwin Hewitt, Karl StrombergGTM026《Algebraic Theories》Ernest G.ManesGTM027《General Topology》John L.Kelley（一般拓扑学）GTM028《Commutative Algebra》VolumeⅠOscar Zariski, Pierre Samuel（交换代数）GTM029《Commutative Algebra》VolumeⅡOscar Zariski, Pierre Samuel（交换代数）GTM030《Lectures in Abstract AlgebraⅠ.Basic Concepts》Nathan Jacobson（抽象代数讲义Ⅰ基本概念分册）GTM031《Lectures in Abstract AlgebraⅡ.Linear Algabra》Nathan.Jacobson（抽象代数讲义Ⅱ线性代数分册）GTM032《Lectures in Abstract AlgebraⅢ.Theory of Fields and Galois Theory》Nathan.Jacobson（抽象代数讲义Ⅲ域和伽罗瓦理论）GTM033《Differential Topology》Morris W.Hirsch（微分拓扑）GTM034《Principles of Random Walk》Frank Spitzer（2ed.）（随机游动原理）GTM035《Several Complex Variables and Banach Algebras》Herbert Alexander, John Wermer（多复变和Banach代数）GTM036《Linear Topological Spaces》John L.Kelley, Isaac Namioka（线性拓扑空间）GTM037《Mathematical Logic》J.Donald Monk（数理逻辑）GTM038《Several Complex Variables》H.Grauert, K.FritzsheGTM039《An Invitation to C*-Algebras》William Arveson（C*-代数引论）GTM040《Denumerable Markov Chains》John G.Kemeny, urie Snell, Anthony W.KnappGTM041《Modular Functions and Dirichlet Series in Number Theory》Tom M.Apostol （数论中的模函数和Dirichlet序列）GTM042《Linear Representations of Finite Groups》Jean-Pierre Serre（有限群的线性表示）GTM043《Rings of Continuous Functions》Leonard Gillman, Meyer JerisonGTM044《Elementary Algebraic Geometry》Keith KendigGTM045《Probability TheoryⅠ》M.Loève（概率论Ⅰ）（4ed.）GTM046《Probability TheoryⅡ》M.Loève（概率论Ⅱ）（4ed.）GTM047《Geometric Topology in Dimensions 2 and 3》Edwin E.MoiseGTM048《General Relativity for Mathematicians》Rainer.K.Sachs, H.Wu伍鸿熙（为数学家写的广义相对论）GTM049《Linear Geometry》K.W.Gruenberg, A.J.Weir（2ed.）GTM050《Fermat's Last Theorem》Harold M.EdwardsGTM051《A Course in Differential Geometry》Wilhelm Klingenberg（微分几何教程）GTM052《Algebraic Geometry》Robin Hartshorne（代数几何）GTM053《A Course in Mathematical Logic for Mathematicians》Yu.I.Manin（2ed.）GTM054《Combinatorics with Emphasis on the Theory of Graphs》Jack E.Graver, Mark E.WatkinsGTM055《Introduction to Operator TheoryⅠ》Arlen Brown, Carl PearcyGTM056《Algebraic Topology：An Introduction》W.S.MasseyGTM057《Introduction to Knot Theory》Richard.H.Crowell, Ralph.H.FoxGTM058《p-adic Numbers, p-adic Analysis, and Zeta-Functions》Neal Koblitz（p-adic 数、p-adic分析和Z函数）GTM059《Cyclotomic Fields》Serge LangGTM060《Mathematical Methods of Classical Mechanics》V.I.Arnold（经典力学的数学方法）（2ed.）GTM061《Elements of Homotopy Theory》George W.Whitehead（同论论基础）GTM062《Fundamentals of the Theory of Groups》M.I.Kargapolov, Ju.I.Merzljakov GTM063《Modern Graph Theory》Béla BollobásGTM064《Fourier Series：A Modern Introduction》VolumeⅠ（2ed.）R.E.Edwards（傅里叶级数）GTM065《Differential Analysis on Complex Manifolds》Raymond O.Wells, Jr.（3ed.）GTM066《Introduction to Affine Group Schemes》William C.Waterhouse（仿射群概型引论）GTM067《Local Fields》Jean-Pierre Serre（局部域）GTM069《Cyclotomic FieldsⅠandⅡ》Serge LangGTM070《Singular Homology Theory》William S.MasseyGTM071《Riemann Surfaces》Herschel M.Farkas, Irwin Kra（黎曼曲面）GTM072《Classical Topology and Combinatorial Group Theory》John Stillwell（经典拓扑和组合群论）GTM073《Algebra》Thomas W.Hungerford（代数）GTM074《Multiplicative Number Theory》Harold Davenport（乘法数论）（3ed.）GTM075《Basic Theory of Algebraic Groups and Lie Algebras》G.P.HochschildGTM076《Algebraic Geometry：An Introduction to Birational Geometry of Algebraic Varieties》Shigeru IitakaGTM077《Lectures on the Theory of Algebraic Numbers》Erich HeckeGTM078《A Course in Universal Algebra》Stanley Burris, H.P.Sankappanavar（泛代数教程）GTM079《An Introduction to Ergodic Theory》Peter Walters（遍历性理论引论）GTM080《A Course in_the Theory of Groups》Derek J.S.RobinsonGTM081《Lectures on Riemann Surfaces》Otto ForsterGTM082《Differential Forms in Algebraic Topology》Raoul Bott, Loring W.Tu（代数拓扑中的微分形式）GTM083《Introduction to Cyclotomic Fields》Lawrence C.Washington（割圆域引论）GTM084《A Classical Introduction to Modern Number Theory》Kenneth Ireland, Michael Rosen（现代数论经典引论）GTM085《Fourier Series A Modern Introduction》Volume 1（2ed.）R.E.Edwards GTM086《Introduction to Coding Theory》J.H.van Lint（3ed .）GTM087《Cohomology of Groups》Kenneth S.Brown（上同调群）GTM088《Associative Algebras》Richard S.PierceGTM089《Introduction to Algebraic and Abelian Functions》Serge Lang（代数和交换函数引论）GTM090《An Introduction to Convex Polytopes》Ame BrondstedGTM091《The Geometry of Discrete Groups》Alan F.BeardonGTM092《Sequences and Series in BanachSpaces》Joseph DiestelGTM093《Modern Geometry-Methods and Applications》（PartⅠ.The of geometry Surfaces Transformation Groups and Fields）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov （现代几何学方法和应用）GTM094《Foundations of Differentiable Manifolds and Lie Groups》Frank W.Warner（可微流形和李群基础）GTM095《Probability》A.N.Shiryaev（2ed.）GTM096《A Course in Functional Analysis》John B.Conway（泛函分析教程）GTM097《Introduction to Elliptic Curves and Modular Forms》Neal Koblitz（椭圆曲线和模形式引论）GTM098《Representations of Compact Lie Groups》Theodor Breöcker, Tammo tom DieckGTM099《Finite Reflection Groups》L.C.Grove, C.T.Benson（2ed.）GTM100《Harmonic Analysis on Semigroups》Christensen Berg, Jens Peter Reus Christensen, Paul ResselGTM101《Galois Theory》Harold M.Edwards（伽罗瓦理论）GTM102《Lie Groups, Lie Algebras, and Their Representation》V.S.Varadarajan（李群、李代数及其表示）GTM103《Complex Analysis》Serge LangGTM104《Modern Geometry-Methods and Applications》（PartⅡ.Geometry and Topology of Manifolds）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov（现代几何学方法和应用）GTM105《SL₂ (R)》Serge Lang（SL₂ (R)群）GTM106《The Arithmetic of Elliptic Curves》Joseph H.Silverman（椭圆曲线的算术理论）GTM107《Applications of Lie Groups to Differential Equations》Peter J.Olver（李群在微分方程中的应用）GTM108《Holomorphic Functions and Integral Representations in Several Complex Variables》R.Michael RangeGTM109《Univalent Functions and Teichmueller Spaces》Lehto OlliGTM110《Algebraic Number Theory》Serge Lang（代数数论）GTM111《Elliptic Curves》Dale Husemoeller（椭圆曲线）GTM112《Elliptic Functions》Serge Lang（椭圆函数）GTM113《Brownian Motion and Stochastic Calculus》Ioannis Karatzas, Steven E.Shreve （布朗运动和随机计算）GTM114《A Course in Number Theory and Cryptography》Neal Koblitz（数论和密码学教程）GTM115《Differential Geometry：Manifolds, Curves, and Surfaces》M.Berger, B.Gostiaux GTM116《Measure and Integral》Volume1 John L.Kelley, T.P.SrinivasanGTM117《Algebraic Groups and Class Fields》Jean-Pierre Serre（代数群和类域）GTM118《Analysis Now》Gert K.Pedersen（现代分析）GTM119《An introduction to Algebraic Topology》Jossph J.Rotman（代数拓扑导论）GTM120《Weakly Differentiable Functions》William P.Ziemer（弱可微函数）GTM121《Cyclotomic Fields》Serge LangGTM122《Theory of Complex Functions》Reinhold RemmertGTM123《Numbers》H.-D.Ebbinghaus, H.Hermes, F.Hirzebruch, M.Koecher, K.Mainzer, J.Neukirch, A.Prestel, R.Remmert（2ed.）GTM124《Modern Geometry-Methods and Applications》（PartⅢ.Introduction to Homology Theory）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov（现代几何学方法和应用）GTM125《Complex Variables：An introduction》Garlos A.Berenstein, Roger Gay GTM126《Linear Algebraic Groups》Armand Borel（线性代数群）GTM127《A Basic Course in Algebraic Topology》William S.Massey（代数拓扑基础教程）GTM128《Partial Differential Equations》Jeffrey RauchGTM129《Representation Theory：A First Course》William Fulton, Joe HarrisGTM130《Tensor Geometry》C.T.J.Dodson, T.Poston（张量几何）GTM131《A First Course in Noncommutative Rings》m（非交换环初级教程）GTM132《Iteration of Rational Functions：Complex Analytic Dynamical Systems》AlanF.Beardon（有理函数的迭代：复解析动力系统）GTM133《Algebraic Geometry：A First Course》Joe Harris（代数几何）GTM134《Coding and Information Theory》Steven RomanGTM135《Advanced Linear Algebra》Steven RomanGTM136《Algebra：An Approach via Module Theory》William A.Adkins, Steven H.WeintraubGTM137《Harmonic Function Theory》Sheldon Axler, Paul Bourdon, Wade Ramey（调和函数理论）GTM138《A Course in Computational Algebraic Number Theory》Henri Cohen（计算代数数论教程）GTM139《Topology and Geometry》Glen E.BredonGTM140《Optima and Equilibria：An Introduction to Nonlinear Analysis》Jean-Pierre AubinGTM141《A Computational Approach to Commutative Algebra》Gröbner Bases, Thomas Becker, Volker Weispfenning, Heinz KredelGTM142《Real and Functional Analysis》Serge Lang（3ed.）GTM143《Measure Theory》J.L.DoobGTM144《Noncommutative Algebra》Benson Farb, R.Keith DennisGTM145《Homology Theory：An Introduction to Algebraic Topology》James W.Vick（同调论：代数拓扑简介）GTM146《Computability：A Mathematical Sketchbook》Douglas S.BridgesGTM147《Algebraic K-Theory and Its Applications》Jonathan Rosenberg（代数K理论及其应用）GTM148《An Introduction to the Theory of Groups》Joseph J.Rotman（群论入门）GTM149《Foundations of Hyperbolic Manifolds》John G.Ratcliffe（双曲流形基础）GTM150《Commutative Algebra with a view toward Algebraic Geometry》David EisenbudGTM151《Advanced Topics in the Arithmetic of Elliptic Curves》Joseph H.Silverman（椭圆曲线的算术高级选题）GTM152《Lectures on Polytopes》Günter M.ZieglerGTM153《Algebraic Topology：A First Course》William Fulton（代数拓扑）GTM154《An introduction to Analysis》Arlen Brown, Carl PearcyGTM155《Quantum Groups》Christian Kassel（量子群）GTM156《Classical Descriptive Set Theory》Alexander S.KechrisGTM157《Integration and Probability》Paul MalliavinGTM158《Field theory》Steven Roman（2ed.）GTM159《Functions of One Complex Variable VolⅡ》John B.ConwayGTM160《Differential and Riemannian Manifolds》Serge Lang（微分流形和黎曼流形）GTM161《Polynomials and Polynomial Inequalities》Peter Borwein, Tamás Erdélyi（多项式和多项式不等式）GTM162《Groups and Representations》J.L.Alperin, Rowen B.Bell（群及其表示）GTM163《Permutation Groups》John D.Dixon, Brian Mortime rGTM164《Additive Number Theory：The Classical Bases》Melvyn B.NathansonGTM165《Additive Number Theory：Inverse Problems and the Geometry of Sumsets》Melvyn B.NathansonGTM166《Differential Geometry：Cartan's Generalization of Klein's Erlangen Program》R.W.SharpeGTM167《Field and Galois Theory》Patrick MorandiGTM168《Combinatorial Convexity and Algebraic Geometry》Günter Ewald（组合凸面体和代数几何）GTM169《Matrix Analysis》Rajendra BhatiaGTM170《Sheaf Theory》Glen E.Bredon（2ed.）GTM171《Riemannian Geometry》Peter Petersen（黎曼几何）GTM172《Classical Topics in Complex Function Theory》Reinhold RemmertGTM173《Graph Theory》Reinhard Diestel（图论）（3ed.）GTM174《Foundations of Real and Abstract Analysis》Douglas S.Bridges（实分析和抽象分析基础）GTM175《An Introduction to Knot Theory》W.B.Raymond LickorishGTM176《Riemannian Manifolds：An Introduction to Curvature》John M.LeeGTM177《Analytic Number Theory》Donald J.Newman（解析数论）GTM178《Nonsmooth Analysis and Control Theory》F.H.clarke, Yu.S.Ledyaev, R.J.Stern, P.R.Wolenski（非光滑分析和控制论）GTM179《Banach Algebra Techniques in Operator Theory》Ronald G.Douglas（2ed.）GTM180《A Course on Borel Sets》S.M.Srivastava（Borel 集教程）GTM181《Numerical Analysis》Rainer KressGTM182《Ordinary Differential Equations》Wolfgang WalterGTM183《An introduction to Banach Spaces》Robert E.MegginsonGTM184《Modern Graph Theory》Béla Bollobás（现代图论）GTM185《Using Algebraic Geomety》David A.Cox, John Little, Donal O’Shea（应用代数几何）GTM186《Fourier Analysis on Number Fields》Dinakar Ramakrishnan, Robert J.Valenza GTM187《Moduli of Curves》Joe Harris, Ian Morrison（曲线模）GTM188《Lectures on the Hyperreals：An Introduction to Nonstandard Analysis》Robert GoldblattGTM189《Lectures on Modules and Rings》m（模和环讲义）GTM190《Problems in Algebraic Number Theory》M.Ram Murty, Jody Esmonde（代数数论中的问题）GTM191《Fundamentals of Differential Geometry》Serge Lang（微分几何基础）GTM192《Elements of Functional Analysis》Francis Hirsch, Gilles LacombeGTM193《Advanced Topics in Computational Number Theory》Henri CohenGTM194《One-Parameter Semigroups for Linear Evolution Equations》Klaus-Jochen Engel, Rainer Nagel（线性发展方程的单参数半群）GTM195《Elementary Methods in Number Theory》Melvyn B.Nathanson（数论中的基本方法）GTM196《Basic Homological Algebra》M.Scott OsborneGTM197《The Geometry of Schemes》David Eisenbud, Joe HarrisGTM198《A Course in p-adic Analysis》Alain M.RobertGTM199《Theory of Bergman Spaces》Hakan Hedenmalm, Boris Korenblum, Kehe Zhu（Bergman空间理论）GTM200《An Introduction to Riemann-Finsler Geometry》D.Bao, S.-S.Chern, Z.Shen GTM201《Diophantine Geometry An Introduction》Marc Hindry, Joseph H.Silverman GTM202《Introduction to Topological Manifolds》John M.LeeGTM203《The Symmetric Group》Bruce E.SaganGTM204《Galois Theory》Jean-Pierre EscofierGTM205《Rational Homotopy Theory》Yves Félix, Stephen Halperin, Jean-Claude Thomas（有理同伦论）GTM206《Problems in Analytic Number Theory》M.Ram MurtyGTM207《Algebraic Graph Theory》Chris Godsil, Gordon Royle（代数图论）GTM208《Analysis for Applied Mathematics》Ward CheneyGTM209《A Short Course on Spectral Theory》William Arveson（谱理论简明教程）GTM210《Number Theory in Function Fields》Michael RosenGTM211《Algebra》Serge Lang（代数）GTM212《Lectures on Discrete Geometry》Jiri Matousek（离散几何讲义）GTM213《From Holomorphic Functions to Complex Manifolds》Klaus Fritzsche, Hans Grauert（从正则函数到复流形）GTM214《Partial Differential Equations》Jüergen Jost（偏微分方程）GTM215《Algebraic Functions and Projective Curves》David M.Goldschmidt（代数函数和投影曲线）GTM216《Matrices：Theory and Applications》Denis Serre（矩阵：理论及应用）GTM217《Model Theory An Introduction》David Marker（模型论引论）GTM218《Introduction to Smooth Manifolds》John M.Lee（光滑流形引论）GTM219《The Arithmetic of Hyperbolic 3-Manifolds》Colin Maclachlan, Alan W.Reid GTM220《Smooth Manifolds and Observables》Jet Nestruev（光滑流形和直观）GTM221《Convex Polytopes》Branko GrüenbaumGTM222《Lie Groups, Lie Algebras, and Representations》Brian C.Hall（李群、李代数和表示）GTM223《Fourier Analysis and its Applications》Anders Vretblad（傅立叶分析及其应用）GTM224《Metric Structures in Differential Geometry》Gerard Walschap（微分几何中的度量结构）GTM225《Lie Groups》Daniel Bump（李群）GTM226《Spaces of Holomorphic Functions in the Unit Ball》Kehe Zhu（单位球内的全纯函数空间）GTM227《Combinatorial Commutative Algebra》Ezra Miller, Bernd Sturmfels（组合交换代数）GTM228《A First Course in Modular Forms》Fred Diamond, Jerry Shurman（模形式初级教程）GTM229《The Geometry of Syzygies》David Eisenbud（合冲几何）GTM230《An Introduction to Markov Processes》Daniel W.Stroock（马尔可夫过程引论）GTM231《Combinatorics of Coxeter Groups》Anders Bjröner, Francesco Brenti（Coxeter 群的组合学）GTM232《An Introduction to Number Theory》Graham Everest, Thomas Ward（数论入门）GTM233《Topics in Banach Space Theory》Fenando Albiac, Nigel J.Kalton（Banach空间理论选题）GTM234《Analysis and Probability：Wavelets, Signals, Fractals》Palle E.T.Jorgensen（分析与概率）GTM235《Compact Lie Groups》Mark R.Sepanski（紧致李群）GTM236《Bounded Analytic Functions》John B.Garnett（有界解析函数）GTM237《An Introduction to Operators on the Hardy-Hilbert Space》Rubén A.Martínez-Avendano, Peter Rosenthal（哈代-希尔伯特空间算子引论）GTM238《A Course in Enumeration》Martin Aigner（枚举教程）GTM239《Number Theory：VolumeⅠTools and Diophantine Equations》Henri Cohen GTM240《Number Theory：VolumeⅡAnalytic and Modern Tools》Henri Cohen GTM241《The Arithmetic of Dynamical Systems》Joseph H.SilvermanGTM242《Abstract Algebra》Pierre Antoine Grillet（抽象代数）GTM243《Topological Methods in Group Theory》Ross GeogheganGTM244《Graph Theory》J.A.Bondy, U.S.R.MurtyGTM245《Complex Analysis：In the Spirit of Lipman Bers》Jane P.Gilman, Irwin Kra, Rubi E.RodriguezGTM246《A Course in Commutative Banach Algebras》Eberhard KaniuthGTM247《Braid Groups》Christian Kassel, Vladimir TuraevGTM248《Buildings Theory and Applications》Peter Abramenko, Kenneth S.Brown GTM249《Classical Fourier Analysis》Loukas Grafakos（经典傅里叶分析）GTM250《Modern Fourier Analysis》Loukas Grafakos（现代傅里叶分析）GTM251《The Finite Simple Groups》Robert A.WilsonGTM252《Distributions and Operators》Gerd GrubbGTM253《Elementary Functional Analysis》Barbara D.MacCluerGTM254《Algebraic Function Fields and Codes》Henning StichtenothGTM255《Symmetry Representations and Invariants》Roe Goodman, Nolan R.Wallach GTM256《A Course in Commutative Algebra》Kemper GregorGTM257《Deformation Theory》Robin HartshorneGTM258《Foundation of Optimization》Osman GülerGTM259《Ergodic Theory：with a view towards Number Theory》Manfred Einsiedler, Thomas WardGTM260《Monomial Ideals》Jurgen Herzog, Takayuki HibiGTM261《Probability and Stochastics》Erhan CinlarGTM262《Essentials of Integration Theory for Analysis》Daniel W.StroockGTM263《Analysis on Fock Spaces》Kehe ZhuGTM264《Functional Analysis, Calculus of Variations and Optimal Control》Francis ClarkeGTM265《Unbounded Self-adjoint Operatorson Hilbert Space》Konrad Schmüdgen GTM266《Calculus Without Derivatives》Jean-Paul PenotGTM267《Quantum Theory for Mathematicians》Brian C.HallGTM268《Geometric Analysis of the Bergman Kernel and Metric》Steven G.Krantz GTM269《Locally Convex Spaces》M.Scott Osborne。
法国数学家拉格朗日著作《解析函数论》英文名
法国数学家拉格朗日著作《解析函数论》英文名全文共3篇示例，供读者参考篇1Title: French Mathematician Lagrange's Work "Analytical Function Theory"Introduction:Lagrange's "Analytical Function Theory" is a seminal work by the French mathematician Joseph-Louis Lagrange, also known as the Lagrange interpolation or Lagrange polynomial. In this work, Lagrange presents a detailed analysis of functions and their properties, laying the foundation for modern function theory. The book delves into topics such as series, limits, derivatives, and integrals of functions, providing a comprehensive study of mathematical functions.Chapter 1: Historical BackgroundJoseph-Louis Lagrange was born in Turin, Italy, in 1736 and later moved to Paris, where he made significant contributions to mathematics, mechanics, and astronomy. Lagrange's work in function theory was influenced by earlier mathematicians such as Euler, d'Alembert, and Legendre. His innovative approach toanalyzing functions set him apart as a pioneering figure in the field of mathematics.Chapter 2: Analytical Function TheoryIn "Analytical Function Theory," Lagrange explores the properties of functions through the use of calculus and algebraic techniques. He introduces the concept of series as a way to represent functions as infinite sums of terms, allowing for a more precise analysis of their behavior. Lagrange also discusses the importance of limits in determining the behavior of functions at particular points, laying the groundwork for modern calculus.Chapter 3: Applications and ImpactLagrange's work on function theory has had a lasting impact on mathematics, with his ideas forming the basis for modern theories in analysis and calculus. The concept of the Lagrange interpolation polynomial, named in his honor, is still widely used in numerical analysis and approximation techniques. His work has inspired generations of mathematicians to further explore the depths of function theory and its applications in various fields.Conclusion:In conclusion, Joseph-Louis Lagrange's work "Analytical Function Theory" stands as a testament to his brilliance and innovative contributions to the field of mathematics. Through his meticulous analysis of functions and their properties, Lagrange paved the way for future developments in function theory and calculus. His work remains a cornerstone of modern mathematics, continuing to inspire mathematicians worldwide to push the boundaries of knowledge in this dynamic field.篇2Title: French Mathematician Lagrange's Work "Analytic Functions Theory"IntroductionLagrange's work on Analytic Functions Theory is a significant contribution to the field of mathematics. This book, also known as "Théorie des Fonctions Analytiques" in French, was published in the year 1797 by the renowned French mathematician Joseph Louis Lagrange. In this seminal work, Lagrange established the foundation for the study of analytic functions and laid the groundwork for the development of complex analysis.Background of LagrangeJoseph Louis Lagrange, born in Turin, Italy in 1736, was a prominent mathematician who made groundbreaking contributions to various fields of mathematics, such as number theory, calculus, and celestial mechanics. He is widely regarded as one of the greatest mathematicians of all time and his work continues to influence modern mathematics.Content of the BookIn "Analytic Functions Theory", Lagrange delves into the study of functions that can be represented by a power series expansion. He introduces key concepts and theorems related to complex analysis, such as Cauchy's integral theorem, the residue theorem, and the fundamental theorem of algebra. Lagrange's work on the properties and behavior of analytic functions revolutionized the field of mathematics and paved the way for further developments in the study of complex numbers.Significance of the WorkLagrange's book is considered a seminal work in the field of mathematics and remains a cornerstone of complex analysis. His contributions to analytic functions theory have had a lasting impact on the field of mathematics and continue to influence contemporary research in areas such as number theory, physics,and engineering. The book has been widely studied and referenced by mathematicians and scientists around the world.ConclusionIn conclusion, Lagrange's work on Analytic Functions Theory stands as a testament to his brilliance and innovation in the field of mathematics. His insights and discoveries continue to shape our understanding of complex analysis and pave the way for new advancements in the field. The book remains a timeless classic in the realm of mathematics and serves as a source of inspiration for generations of mathematicians to come.篇3Title: The Analytic Function Theory by French Mathematician LagrangeIntroduction:Joseph-Louis Lagrange, a renowned mathematician from France, made significant contributions to the field of mathematics during the 18th century. One of his most influential works is the book "Analytic Function Theory," where he laid down the foundations for the study of complex functions. In this article, we will delve into the contents of this seminal work and discuss its impact on the development of mathematics.Overview of the Book:Lagrange's "Analytic Function Theory" is a comprehensive treatise on the analysis of complex functions, which play a crucial role in a variety of mathematical disciplines including calculus, differential equations, and number theory. The book is divided into several sections, each covering different aspects of the theory of analytic functions. Lagrange begins by introducing the basic concepts of complex numbers and functions, before delving into more advanced topics such as power series, contour integration, and the Cauchy-Riemann equations.Key Concepts and Theorems:One of the key contributions of Lagrange in this work is the development of the Cauchy Integral Formula, which provides a powerful method for calculating complex integrals over closed curves. This formula has important applications in the study of harmonic functions and the theory of residues. Lagrange also proved several important theorems in the book, including the Maximum Modulus Principle and the Riemann Mapping Theorem, which have been instrumental in the development of complex analysis.Impact on Mathematics:Lagrange's "Analytic Function Theory" is considered a seminal work in the field of complex analysis and has had a lasting impact on the development of mathematics. The insights and techniques introduced by Lagrange in this book have been instrumental in solving many mathematical problems in diverse areas such as physics, engineering, and computer science. The book continues to be studied and referenced by mathematicians and researchers around the world, highlighting the enduring legacy of Lagrange's contributions to the field.Conclusion:In conclusion, Joseph-Louis Lagrange's "Analytic Function Theory" stands as a cornerstone in the field of complex analysis, providing a solid foundation for the study of analytic functions and their applications. The insights and theorems introduced by Lagrange in this book have had a profound impact on the development of mathematics, shaping the way we understand and solve complex mathematical problems. As we continue to push the boundaries of mathematical research, Lagrange's work remains as relevant and influential as ever.。
2024年AP Calculus AB真题全方位解读
2024年AP Calculus AB真题全方位解读敬爱的读者朋友们，今天我将为大家带来对2024年AP Calculus AB真题的全方位解读。
Calculus AB作为高中阶段的高等数学科目，是一门考察学生对微积分概念、方法和应用的综合能力的考试。
通过对本次考试题目的详细解析，我将帮助大家更好地理解这门学科的要点和考试要求。
本次考试共分为两个部分：选择题和解答题。
以下将从宏观和微观两个层面对每个题目进行解析，帮助读者朋友们全面掌握考试要点。
一、选择题解析选择题是考察学生对概念理解和快速计算的能力的重要方式。
以下是本次考试中的几道典型选择题：1. 在区间[0, π] 上考虑函数 f(x) = sinx + cosx，那么 f'(x) 的值在以下哪个区间上均为正数？A. (0, π/2)B. (π/2, π)C. (π/4, 3π/4)D. (3π/4, π)解析：我们首先求出 f'(x) 的导函数为 f'(x) = cosx - sinx。
然后，我们通过观察 sinx 和 cosx 在[0, π] 区间上的图像变化，可以得知当π/4 < x < 3π/4 时，cosx - sinx 大于零，故选项 C 正确。
2. 函数 f(x) = x^3 - 5x^2 + 2x 的图像在以下哪个点处有一个局部极小值？A. (0, 0)B. (1, -1)C. (1, 2)D. (2, -4)解析：我们首先求出 f'(x) 的导函数为 f'(x) = 3x^2 - 10x + 2。
然后，我们将导函数的根代入 f(x) 中，计算函数值。
经计算可知，当 x = 1 时，f(x) 取得局部极小值，故选项 C 正确。
通过以上两个例题的解析，我们可以看出选择题在考察学生对概念和方法的理解上具有一定的难度。
但只要我们掌握了基本的概念和计算方法，就能够迎接这些挑战。
二、解答题解析解答题是考察学生对概念和计算方法的深入理解以及解决实际问题的能力。
GammaFunction：伽玛函数
Search Site12,720 entries Calculus and Analysis > Special Functions > Gamma Functions Calculus and Analysis > Special Functions > Named Integrals Calculus and Analysis > Special Functions > Product Functions Gamma FunctionThe (complete) gamma function is defined to be an extension of the factorial to arguments. It is related to the factorial bya slightly unfortunate notation due to Legendre which is now universally used instead of Gauss's simpler (Gauss 1812; Edwards 2001, p. 8). everywhere except at, -1, -2, ..., and the residue atisThere are no points at which.There are a number of notational conventions in common use for indication of a power of a gamma functions. While authors such as Watson (1939) use (i.e., using a trigonometric function-like convention), it is also common to write . for(Euler's integral form)The complete gamma function can be generalized to the upper and lower incomplete gamma function .Min MaxRe Register for Unlimited Interactive Examples >>Im ReplotPlots of the real and imaginary parts of in the complex plane are illustrated above.Wolfram Research Demonstrations Site Integrator Tones Functions Site Wolfram Science more…Download Complete Documentation >>Find 171 formulas about the Gamma Function at -55-55(8)(9)If is an integer, 2, 3, ..., then(10)(11)so the gamma function reduces to the factorial for a positive integer argument.A beautiful relationship betweenand the Riemann zeta functionis given by(12)for(Havil 2003, p. 60).The gamma function can also be defined by an infinite product form (Weierstrass form )(13)where is the Euler-Mascheroni constant (Krantz 1999, p. 157; Havil 2003, p. 57). This can be written(14)where(15)(16)for, where is the Riemann zeta function (Finch 2003). Taking the logarithm of both sidesof (◇),(17)Differentiating,(18)(19)(20)(21)(22)(23)(24)(25)where is the digamma function and is the polygamma function . th derivatives are given in terms of the polygamma functions , , ..., .The minimum valueoffor real positiveis achieved when(26)(27)This can be solved numerically to give (Sloane's A030169; Wrench 1968), which has continued fraction [1, 2, 6, 63, 135, 1, 1, 1, 1, 4, 1, 38, ...] (Sloane's A030170). At ,achieves the value 0.8856031944... (Sloane's A030171), which has continued fraction [0, 1, 7, 1, 2, 1, 6, 1, 1, ...] (Sloane's A030172).The Euler limit form is(28)so(29)(30)(31)(32)(Krantz 1999, p. 156).One over the gamma functionis an entire function and can be expressed as(33)where is the Euler-Mascheroni constant and is the Riemann zeta function (Wrench 1968). An asymptotic series for is given by(34) Writing(35) the satisfy(36) (Bourguet 1883, Davis 1933, Isaacson and Salzer 1943, Wrench 1968). Wrench (1968) numerically computed the coefficients for the series expansion about 0 of(37) The Lanczos approximation gives a series expansion for for in terms of an arbitrary constant such that .The gamma function satisfies the functional equations(38)(39) Additional identities are(40)(41)(42)(43)Using (40), the gamma function of a rational number can be reduced to a constant times or . For example,(44)(45)(46)(47) For ,(48) Gamma functions of argument can be expressed using the Legendre duplication formula(49) Gamma functions of argument can be expressed using a triplication formula(50) The general result is the Gauss multiplication formula(51) The gamma function is also related to the Riemann zeta function by(52) For integer , 2, ..., the first few values of are 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, ... (Sloane's A000142). For half-integer arguments, has the special form(53)where is a double factorial. The first few values for , 3, 5, ... are therefore(54)(55)(56), , ... (Sloane's A001147 and A000079; Wells 1986, p. 40). In general, for a positive integer, 2, ...(57)(58)(59)(60) Simple closed-form expressions of this type do not appear to exist for for a positive integer . However, Borwein and Zucker (1992) give a variety of identities relating gamma functions to square roots and elliptic integral singular values, i.e., elliptic moduli such that(61)where is a complete elliptic integral of the first kind and is the complementary integral. M. Trott (pers. comm.) has developed an algorithm for automatically generating hundreds of such identities.(62)(63)(64)(65)(66)(67)(68)(69)(70)(71)(72)(73)(74)(75)(76)(77)(78)(79)(80)(81)(82)Several of these are also given in Campbell (1966, p. 31).A few curious identities include(83)(84)(85)(86)(87)(88)(89) of which Magnus and Oberhettinger 1949, p. 1 give only the last case,(90) and(91)(Magnus and Oberhettinger 1949, p. 1). Ramanujan also gave a number of fascinating identities:(92)(93) where(94)(95)(Berndt 1994).Ramanujan gave the infinite sums(96)(97) and(98)(99)(Hardy 1923; Hardy 1924; Whipple 1926; Watson 1931; Bailey 1935; Hardy 1999, p. 7).The following asymptotic series is occasionally useful in probability theory (e.g., theone-dimensional random walk):(100) (Graham et al. 1994). This series also gives a nice asymptotic generalization of Stirling numbers of the first kind to fractional values.It has long been known that is transcendental (Davis 1959), as is (Le Lionnais 1983; Borwein and Bailey 2003, p. 138), and Chudnovsky has apparently recently proved thatis itself transcendental (Borwein and Bailey 2003, p. 138).There exist efficient iterative algorithms for for all integers (Borwein and Bailey 2003, p. 137). For example, a quadratically converging iteration for (Sloane'sA068466) is given by defining(101)(102) setting and , and then(103)(Borwein and Bailey 2003, pp. 137-138).No such iteration is known for (Borwein and Borwein 1987; Borwein and Zucker 1992; Borwein and Bailey 2003, p. 138).SEE ALSO:Bailey's Theorem, Barnes G-Function, Binet's Fibonacci Number Formula, Bohr-Mollerup Theorem, Digamma Function, Double Gamma Function, Fransén-Robinson Constant Gauss Multiplication Formula, Incomplete Gamma Function, Knar's Formula, Lambda Function, LanczosApproximation, Legendre Duplication Formula, Log Gamma Function, Mellin's Formula, Mu Function, Nu Function, Pearson's Function, Polygamma Function, Regularized Gamma Function, Stirling's Series, Superfactorial. [Pages Linking Here]RELATED WOLFRAM SITES:/GammaBetaErf/Gamma/,/GammaBetaErf/LogGamma/REFERENCES:Abramowitz, M. and Stegun, I. A. (Eds.). "Gamma (Factorial) Function" and "Incomplete Gamma Function." §6.1 and 6.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 255-258 and 260-263, 1972.Arfken, G. "The Gamma Function (Factorial Function)." Ch. 10 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 339-341 and 539-572, 1985.Artin, E. The Gamma Function. New York: Holt, Rinehart, and Winston, 1964.Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, 1935.Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 334-342, 1994.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 218, 1987.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Borwein, J. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 6, 1987.Borwein, J. M. and Zucker, I. J. "Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete Elliptic Integrals of the First Kind." IMA J. Numerical Analysis12, 519-526, 1992.Bourguet, L. "Sur les intégrales Eulériennes et quelques autres fonctions uniformes." Acta Math.2, 261-295, 1883. Campbell, R. Les intégrales eulériennes et leurs applications. Paris: Dunod, 1966.Davis, H. T. Tables of the Higher Mathematical Functions. Bloomington, IN: Principia Press, 1933.Davis, P. J. "Leonhard Euler's Integral: A Historical Profile of the Gamma Function." Amer. Math. Monthly66, 849-869, 1959.Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "The Gamma Function." Ch. 1 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 1-55, 1981.Finch, S. R. "Euler-Mascheroni Constant." §1.5 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 28-40, 2003.Gauss, C. F. "Disquisitiones Generales Circa Seriem Infinitametc. Pars Prior." Commentationes Societiones Regiae Scientiarum Gottingensis Recentiores, Vol. II. 1812. Reprinted in Gesammelte Werke, Bd. 3, pp. 123-163 and 207-229, 1866.Graham, R. L.; Knuth, D. E.; and Patashnik, O. Answer to Problem 9.60 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.Hardy, G. H. "A Chapter from Ramanujan's Note-Book." Proc. Cambridge Philos. Soc.21, 492-503, 1923.Hardy, G. H. "Some Formulae of Ramanujan." Proc. London Math. Soc. (Records of Proceedings at Meetings) 22,xii-xiii, 1924.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Havil, J. "The Gamma Function." Ch. 6 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 53-60, 2003.Isaacson, E. and Salzer, H. E. "Mathematical Tables--Errata: 19. J. P. L. Bourget, 'Sur les intégrales Eulériennes et quelques autres fonctions uniformes,' Acta Mathematica, v. 2, 1883, pp. 261-295.' " Math. Tab. Aids Comput.1, 124, 1943.Koepf, W. "The Gamma Function." Ch. 1 in Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 4-10, 1998.Krantz, S. G. "The Gamma and Beta Functions." §13.1 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 155-158, 1999.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 46, 1983.Magnus, W. and Oberhettinger, F. Formulas and Theorems for the Special Functions of Mathematical Physics. New York: Chelsea, 1949.Nielsen, N. "Handbuch der Theorie der Gammafunktion." Part I in Die Gammafunktion. New York: Chelsea, 1965. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Gamma Function, Beta Function, Factorials, Binomial Coefficients" and "Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function." §6.1 and 6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 206-209 and 209-214, 1992.Sloane, N. J. A. Sequences A000079/M1129, A000142/M1675, A001147/M3002, A030169, A030170, A030171,A030172, and A068466 in "The On-Line Encyclopedia of Integer Sequences."Spanier, J. and Oldham, K. B. "The Gamma Function " and "The Incomplete Gamma and Related Functions." Chs. 43 and 45 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 411-421 and 435-443, 1987. Watson, G. N. "Theorems Stated by Ramanujan (XI)." J. London Math. Soc.6, 59-65, 1931.Watson, G. N. "Three Triple Integrals." Quart. J. Math., Oxford Ser. 210, 266-276, 1939.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 40, 1986. Whipple, F. J. W. "A Fundamental Relation between Generalised Hypergeometric Series." J. London Math. Soc.1,138-145, 1926.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.Wrench, J. W. Jr. "Concerning Two Series for the Gamma Function." Math. Comput.22, 617-626, 1968.LAST MODIFIED:December 26, 2005CITE THIS AS:Weisstein, Eric W. "Gamma Function." From MathWorld--A Wolfram Web Resource./GammaFunction.html© 1999 CRC Press LLC, © 1999-2007 Wolfram Research, Inc. | Terms of Use。
08-09(2)Calculus(II)_Exam_Paper(B)参考答案
= −2(m tan m + ln |cos m|) + C (C ∈ R) √ √ = −2 arctan(arcsin 1 − x) arcsin 1 − x + ln √ 2. 交换积分顺序。先对y 积分，再对x积分。 ∫ π ∫ sin x √ 2 Σ = dx cos x 1 + cos2 xdy ∫ = ∫ = =
所以有： x2 z 2 + 2yz y2 = − 2 z + 2xz = − 3
∂y 要求dy ，即求y = y (x, z )的全微分，类似上面求出 ∂x 和 ∂y ∂z （由x， y ， z 的对称性易得） ，所以有：
dy = =
∂y ∂y dx + dz ∂x ∂z −x2 −z 2 dx + dz y 2 + 2yz y 2 + 2yx
0 0 1 −1
∫ ∫
1 dy y2
1 0 1 0 1 0
∫
1
tan y 2 xd(y 2 x)
−1
= − = − ∫ = −
1 dy y2
∫
1
−1
1 d(cos y 2 x) cos y 2 x
1 dy (ln cos y 2 x |1 −1 ) y2 1 dy (ln cos y 2 − ln cos(−y 2 ) ) = 0 y2
2. 易知过(1, e)点的切线斜率为e，所以切线为y − e = e(x − 1)，即y = ex。 积分区域即为由y = ex 、y = 0、y = ex围成的区域。 ∫ ∫ S = dxdy ∫ 1 ∫ ex dy dx = 0 ex ∫ 1 1 = (ex − ex)dx = e − 1 2 0 3. 沿x轴做切面，得到是一个圆环带，外环是ex ，内环是ex，则每个切 面的面积是π [(ex )2 − (ex)2 ]。再沿x轴累加即得体积。 ∫ V =
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1.2 Historical Survey Most authors on this topic will cite a particular date as the birthday of so called 'Fractional Calculus'. In a letter dated September 30th, 1695 L’Hospital wrote to Leibniz asking him about a particular notation he had used in his publications for the nth-derivative of the Dn x linear function f(x) = x, . L’Hospital posed the question to Leibniz, Dx n what would the result be if n = 1/ 2. Leibniz's response: "An apparent paradox, from which one day useful consequences will be drawn." In these words fractional calculus was born. [7]
Chapter 1
Definition and Applications
3
Fractional Calculus History
value x, verifiable by infinite series expansion, or more practically, by calculator. Now, in the same way consider the integral and derivative. Although they are indeed concepts of a higher complexity by nature, it is still fairly easy to physically represent their meaning. Once mastered, the idea of completing numerous of these operations, integrations or differentiations follows naturally. Given the satisfaction of a very few restrictions (e.g. function continuity) completing n integrations can become as methodical as multiplication. [7] But the curious mind can not be restrained from asking the question what if n were not restricted to an integer value? Again, at first glance, the physical meaning can become convoluted (pun intended), but as this report will show, fractional calculus flows quite naturally from our traditional definitions. And just as fractional exponents such as the square root may find their way into innumerable equations and applications, it will become apparent that integrations of order 1/ 2 and beyond can find practical use in many modern problems. [7]
and the integration operator J. (Usually J is used in favor of I to avoid confusion with other I-like glyphs and identities) [1] Fractional calculus comes out of generalizing the differential/integral operator, applying it at non-integer order (example, a "half derivative" operation such that taking the half derivative of a function twice yields the first derivative The traditional integral and derivative are, to say the least, a staple for the technology professional, essential as a means of understanding and working with natural and artificial systems. [7] Fractional Calculus is a field of mathematic study that grows out of the traditional definitions of the calculus integral and derivative operators in much the same way fractional exponents is an outgrowth of exponents with integer value. [7]
Chapter 1
Definition and Applications
2
Fractional Calculus History
1.1
Definition
Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers of the differential operators
Chapter 1
Definition and Applilus History
Leibniz
Following L’Hospital and Leibniz’s first inquisition, fractional calculus was primarily a study reserved for the best minds in mathematics. Fourier, Euler, Laplace are among the many that dabbled with fractional calculus and the mathematical consequences. Many found, using their own notation and methodology, definitions that fit the concept of a non-integer order integral or derivative. [7] The most famous of these definitions that have been popularized in the world of fractional calculus (not yet the world as a whole) are the Riemann-Liouville and Grunwald-Letnikov definition. While the shear numbers of actual Definitions are no doubt as numerous as the men and women that study this field, they are for the most part variations on the themes of these two and so are addressed in detail in this document. [7] Most of the mathematical theory applicable to the study of fractional calculus was developed prior to the turn of the 20th century. However it is in the past 100 years that the most intriguing leaps in engineering and scientific application have been found. The
Chapter 1
Definition and Applications
5
Fractional Calculus History
mathematics has in some cases had to change to meet the requirements of physical reality. [7] Caputo reformulated the more 'classic' definition of the RiemannLiouville fractional derivative in order to use integer order initial conditions to solve his fractional order differential equations. As recently as 1996, Kolowankar reformulated again, the RiemannLiouville fractional derivative in order to differentiate no-where differentiable fractal functions. [7] Leibniz's response, based on studies over the intervening 300 years, has proven at least half right. It is clear that within the 20th century especially numerous applications and physical manifestations of fractional calculus have been found. However, these applications and the mathematical background surrounding fractional calculus are far from paradoxical. While the physical meaning is difficult (arguably impossible) to grasp, the definitions themselves are no more rigorous than those of their integer order counterparts. [7] Understanding of definitions and use of fractional calculus will be made clearer by quickly discussing some necessary but relatively simple mathematical definitions that will arise in the study of these concepts. These are The Gamma Function, The Beta Function, The Laplace Transform, and the Mittag-Leffler Function and are addressed in the second and third chapter of this thesis. [7]