Introduction to calculus and analysis / Richard Courant and Fritz John.
Editor: New York : Interscience Publishers, c1965-c1974Descripción: 2 v. (661, 954 p.) : il. ; 24 cmISBN: 0471178624 (v. 2)Tema(s): Calculus | Mathematical analysisOtra clasificación: 00A05 (26-01)Chapter 1 Introduction [1] 1.1 The Continuum of Numbers [1] a. The System of Natural Numbers and Its Extension. Counting and Measuring, [1] b. Real Numbers and Nested Intervals, [7] c. Decimal Fractions. Bases Other Than Ten, [9] d. Definition of Neighborhood, [12] e. Inequalities, [12] 1.2 The Concept of Function [17] a. Mapping-Graph, [18] b. Definition of the Concept of Functions of a Continuous Variable. Domain and Range of a Function, [21] c. Graphical Representation. Monotonic Functions, [24] d. Continuity, 31 e. The Intermediate Value Theorem. Inverse Functions, [44] 1.3 The Elementary Functions [47] a. Rational Functions, [47] b. Algebraic Functions, [49] c. Trigonometric Functions, [49] d. The Exponential Function and the Logarithm, [51] e. Compound Functions, Symbolic Products, Inverse Functions, [52] 1.4 Sequences [55] 1.5 Mathematical Induction [57] 1.6 The Limit of a Sequence [60] a. an = 1/n, [61] b. a2m = 1/m; a2m-1 = 1/2m, [62] c. an = n/n+1, [63] d. an = n√p , [64] e. an = αn , [65] f. Geometrical Illustration of the Limits of αn and n√p, [65] g. The Geometric Series, [67] h. an = n√n , [69] i. an = (√n+1) - √n, [69] 1.7 Further Discussion of the Concept of Limit [70] a. Definition of Convergence and Divergence, [70] b. Rational Operations with Limits, [71] c. Intrinsic Convergence Tests. Monotone Sequences, [73] d. Infinite Series and the Summation Symbol, [75] e. The Number e, [77] f. The Number п as a Limit, [80] 1.8 The Concept of Limit for Functions of a Continuous Variable [82] a. Some Remarks about the Elementary Functions, [86] Supplements [87] 5.1 Limits and the Number Concept [89] a. The Rational Numbers, [89] b. Real Numbers Determined by Nested Sequences of Rational Intervals, [90] c. Order, Limits, and Arithmetic Operations for Real Numbers, [92] d. Completeness of the Number Continuum. Compactness of Closed Intervals. Convergence Criteria, [94] e. Least Upper Bound and Greatest Lower Bound, [97] f. Denumerability of the Rational Numbers, [98] 5.2 Theorems on Continuous Functions [99] 5.3 Polar Coordinates [101] S.4 Remarks on Complex Numbers [103] PROBLEMS [106] Chapter The Fundamental Ideas of the Integral and Differential Calculus [119] 2.1 The Integra] [120] a. Introduction, [120] b. The Integral as an Area, [121] c. Analytic Definition of the Integral. Notations, [122] 2.2 Elementary Examples of Integration [128] a. Integration of Linear Function, [128] b. Integration of x2, [130] c. Integration of xα for Integers α ≠ -1, [131] d. Integration of xα for Rational α Other Than -1, [134] e. Integration of sin x and cos x, [135] 2.3 Fundamental Rules of Integration [136] a. Additivity, [136] b. Integral of a Sum of a Product with a Constant, [137] c. Estimating Integrals, 138, d. The Mean Value Theorem for Integrals, [139] 2.4 The Integral as a Function of the Upper Limit (Indefinite Integral) [143] 2.5 Logarithm Defined by an Integral [145] a. Definition of the Logarithm Function, [145] b. The Addition Theorem for Logarithms, [147] 2.6 Exponential Function and Powers [149] a. The Logarithm of the Number e, [149] b. The Inverse Function of the Logarithm. The Exponential Function, [150] c. The Exponential Function as Limit of Powers, [152] d. Definition of Arbitrary Powers of Positive Numbers, [152] e. Logarithms to Any Base, [153] 2.7 The Integral of an Arbitrary Power of x [154] 2.8 The Derivative [155] a. The Derivative and the Tangent, [156] b. The Derivative as a Velocity, [162] c. Examples of Differentiation, [163] d. Some Fundamental Rules for Differentiation, [165] e. Differentiability and Continuity of Functions, [166] f. Higher Derivatives and Their Significance, [169] g. Derivative and Difference Quotient. Leibnitz’s Notation, [171] h. The Mean Value Theorem of Differential Calculus, [173] i. Proof of the Theorem, [175] j. The Approximation of Functions by Linear Functions. Definition of Differentials, [179] k. Remarks on Applications to the Natural Sciences, [183] 2.9 The Integral, the Primitive Function, and the Fundamental Theorems of the Calculus [184] a. The Derivative of the Integral, [184] b. The Primitive Function and Its Relation to the Integral, [186] c. The Use of the Primitive Function for Evaluation of Definite Integrals, [189] d. Examples, [191] Supplement The Existence of the Definite Integral of a Continuous Function [192] PROBLEMS [196] Chapter 3 The Techniques of Calculus [201] Part A Differentiation and Integration of the Elementary Functions [201] 3.1 The Simplest Rules for Differentiation and Their Applications [201] a. Rules for Differentiation, [201] b. Differentiation of the Rational Functions, [204] c. Differentiation of the Trigonometric Functions, [205] 3.2 The Derivative of the Inverse Function [206] a. General Formula, [206] b. The Inverse of the nth Power; the nth Root, [210] c. The Inverse Trigonometric Functions—Multivaluedness, [210] d. The Corresponding Integral Formulas, [215] e. Derivative and Integral of the Exponential Function, [216] 3.3 Differentiation of Composite Functions [217] a. Definitions, [217] b. The Chain Rule, [218] c. The Generalized Mean Value Theorem of the Differential Calculus, [222] 3.4 Some Applications of the Exponential Function [223] a. Definition of the Exponential Function by Means of a Differential Equation, [223] b. Interest Compounded Continuously. Radioactive Disintegration, [224] c. Cooling or Heating of a Body by a Surrounding Medium, [225] d. Variation of the Atmospheric Pressure with the Height above the Surface of the Earth, [226] e. Progress of a Chemical Reaction, [227] f. Switching an Electric Circuit on or off, [228] 3.5 The Hyperbolic Functions [228] a. Analytical Definition, [228] b. Addition Theorems and Formulas for Differentiation [231] c. The Inverse Hyperbolic Functions, [232] d. Further Analogies, [234] 3.6 Maxima and Minima [236] a. Convexity and Concavity of Curves, [236] b. Maxima and Minima—Relative Extrema. Stationary Points, [238] 3.7 The Order of Magnitude of Functions [248] a. The Concept of Order of Magnitude. The Simplest Cases, [248] b. The Order of Magnitude of the Exponential Function and of the Logarithm, 249 c. General Remarks, [251] d. The Order of Magnitude of a Function in the Neighborhood of an Arbitrary Point, [252] e. The Order of Magnitude (or Smallness) of a Function Tending to Zero, [252] f. The “O” and “o” Notation for Orders of Magnitude, [253] APPENDIX [255] A.1 Some Special Functions [255] a. The Function y = e1/x2 , [255] b. The Function y = e1/x, [256] c. The Function y = tanh 1/x, [257] d. The Function y — x tanh 1/x, [258] e. The Function y - x sin 1/x, y(0) = 0, [259] A.2 Remarks on the Differentiability of Functions [259] Part B Techniques of Integration [261] 3.8 Table of Elementary Integrals [263] 3.9 The Method of Substitution [263] a. The Substitution Formula. Integral of a Composite Function, [263] b. A Second Derivation of the Substitution Formula, [268] c. Examples. Integration Formulas, [270] 3.10 Further Examples of the Substitution Method [271] 3.11 Integration by Parts [274] a. General Formula, [274] b. Further Examples of Integration by Parts, [276] c. Integral Formula for (b) + f(a) [278] d. Recursive Formulas, [278] e. Wallis’s Infinite Product for п, [280] 3.12 Integration of Rational Functions [282] a. The Fundamental Types, [283] b. Integration of the Fundamental Types, [284] e. Partial Fractions, [286] d. Examples of Resolution into Partial Fractions. Method of Undetermined Coefficients, [288] 3.13 Integration of Some Other Classes of Functions [290] a. Preliminary Remarks on the Rational Representation of the Circle and the Hyperbola, [290] b. Integration of R(cos x, sin x), [293] c. Integration of R(cosh x, sinh x), [294] d. Integration of r(x, √(l — x2)), [294] e. Integration of r(x, √(x2— 1)), [295] f. Integration of r(x, √(ax2 + 2bx + c)), [295] g. Integration of R(x, Vox2 + 2bx + c), [295] h. Further Examples of Reduction to Integrals of Rational Functions, [296] i. Remarks on the Examples, [297] Part C Further Steps in the Theory of Integral Calculus [298] 3.14 Integrals of Elementary Functions [298] a. Definition of Functions by Integrals. Elliptic Integrals and Functions, [298] b. On Differentiation and Integration, [300] 3.15 Extension of the Concept of Integral [301] a. Introduction. Definition of “Improper” Integrals, [301] b. Functions with Infinite Discontinuities, [303] c. Interpretation as Areas, 304 d. Tests for Convergence, [305] e. Infinite Interval of Integration, [306] f. The Gamma Function, [308] g. The Dirichlet Integral, [309] h. Substitution. Fresnel Integrals, [310] 3.16 The Differential Equations of the Trigonometric Functions [312] a. Introductory Remarks on Differential Equations, [312] b. Sin x and cos x defined by a Differential Equation and Initial Conditions, [312] PROBLEMS [314]
Chapter 4 Applications in Physics and Geometry [324] 4.1 Theory of Plane Curves [324] a. Parametric Representation, [324] b. Change of Parameters, [326] c. Motion along a Curve. Time as the Parameter. Example of the Cycloid, [328] d. Classifications of Curves. Orientation, [333] e. Derivatives. Tangent and Normal, in Parametric Representation, [343] f. The Length of a Curve, [348] g. The Arc Length as a Parameter, [352] h. Curvature, [354] i. Change of Coordinate Axes. Invariance, [360] j. Uniform Motion in the Special Theory of Relativity, [363] k. Integrals Expressing Area within Closed Curves, [365] l. Center of Mass and Moment of a Curve, [373] m. Area and Volume of a Surface of Revolution, [374] n. Moment of Inertia, [375] 4.2 Examples [376] a. The Common Cycloid, [376] b. The Catenary, [378] c. The Ellipse and the Lemniscate, [378] 4.3 Vectors in Two Dimensions [379] a. Definition of Vectors by Translation. Notations, [380] b. Addition and Multiplication of Vectors, [384] c. Variable Vectors, Their Derivatives, and Integrals, [392] d. Application to Plane Curves. Direction, Speed, and Acceleration, [394] 4.4 Motion of a Particle under Given Forces [397] a. Newton’s Law of Motion, [397] b. Motion of Falling Bodies, [398] c. Motion of a Particle Constrained to a Given Curve, [400] 4.5 Free Fall of a Body Resisted by Air [402] 4.6 The Simplest Type of Elastic Vibration [404] 4.7 Motion on a Given Curve [405] a. The Differential Equation and Its Solution, [405] b. Particle Sliding down a Curve, [407] c. Discussion of the Motion, [409] d. The Ordinary Pendulum, [410] e. The Cycloidal Pendulum, [411] 4.8 Motion in a Gravitational Field [413] a. Newton’s Universal Law of Gravitation, [413] b. Circular Motion about the Center of Attraction, [415] c. Radial Motion—Escape Velocity, [416] 4.9 Work and Energy [418] a. Work Done by Forces during a Motion, [418] b. Work and Kinetic Energy. Conservation of Energy, [420] c. The Mutual Attraction of Two Masses, [421] d. The Stretching of a Spring, [423] e. The Charging of a Condenser, [423] APPENDIX [424] A.l Properties of the Evolute [424] A.2 Areas Bounded by Closed Curves. Indices [430] PROBLEMS [435] Chapter 5 Taylor's Expansion [440] 5.1 Introduction: Power Series [440] 5.2 Expansion of the Logarithm and the Inverse Tangent [442] a. The Logarithm, 442 b. The Inverse Tangent, [444] 5.3 Taylor’s Theorem [445] a. Taylor’s Representation of Polynomials, [445] b. Taylor’s Formula for Nonpolynomial Functions, [446] 5.4 Expression and Estimates for the Remainder [447] a. Cauchy’s and Lagrange’s Expressions, [447] b. An Alternative Derivation of Taylor’s Formula, [450] 5.5 Expansions of the Elementary Functions [453] a. The Exponential Function, [453] b. Expansion of sin x, cos x, sinh x, cosh x, [454] c. The Binomial Series, [456] 5.6 Geometrical Applications [457] a. Contact of Curves, [458] b. On the Theory of Relative Maxima and Minima, [461] APPENDIX I [462] A.I.1 Example of a Function Which Cannot Be Expanded in a Taylor Series [462] A.I.2 Zeros and Infinites of Functions [463] a. Zeros of Order n, 463 b. Infinity of Order v, [463] A.I.3 Indeterminate Expressions [464] A.I.4 The Convergence of the Taylor Series of a Function with Nonnegative Derivatives of all Orders [467] APPENDIX H INTERPOLATION [470] A.II.1 The Problem of Interpolation. Uniqueness [470] A.II.2 Construction of the Solution. Newton’s Interpolation Formula [471] A.n.3 The Estimate of the Remainder [474] A.II.4 The Lagrange Interpolation Formula [476] PROBLEMS [477] Chapter 6 Numerical Methods [481] 6.1 Computation of Integrals [482] a. Approximation by Rectangles, [482] b. Refined Approximations—Simpson’s Rule, [483] 6.2 Other Examples of Numerical Methods [490] a. The “Calculus of Errors”, [490] b. Calculation of п, 492 c. Calculation of Logarithms, [493] 6.3 Numerical Solution of Equations [494] a. Newton’s Method, 495 b. The Rule of False Position, 497 c. The Method of Iteration, [499] d. Iterations and Newton’s Procedure, [502] APPENDIX [504] A.l Stirling’s Formula [504] PROBLEMS [507] Chapter 7 Infinite Sums and Products [510] 7.1 The Concepts of Convergence and Divergence [511] a. Basic Concepts, [511] b. Absolute Convergence and Conditional Convergence, [513] c. Rearrangement of Terms, [517] d. Operations with Infinite Series, [520] 7.2 Tests for Absolute Convergence and Divergence [520] a. The Comparison Test. Majorants, [520] b. Convergence Tested by Comparison with the Geometric Series, [521] c. Comparison with an Integral, [524] 7.3 Sequences of Functions [526] a. Limiting Processes with Functions and Curves, [527] 7.4 Uniform and Nonuniform Convergence [529] a. General Remarks and Definitions, [529] b. A Test for Uniform Convergence, [534] c. Continuity of the Sum of a Uniformly Convergent Series of Continuous Functions, [535] d. Integration of Uniformly Convergent Series, [536] e. Differentiation of Infinite Series, [538] 7.5 Power Series [540] a. Convergence Properties of Power Series— Interval of Convergence, [540] b. Integration and Differentiation of Power Series, [542] c. Operations with Power Series, [543] d. Uniqueness of Expansion, [544] e. Analytic Functions, [545] 7.6 Expansion of Given Functions in Power Series. Method of Undetermined Coefficients. Examples [546] a. The Exponential Function, [546] b. The Binomial Series, [546] c. The Series for arc sin x, [549] d. The Series for ar sinh x = log [x + √(1 + x2)], [549] e. Example of Multiplication of Series, [550] f. Example of Term-by-Term Integration (Elliptic Integral), [550] 7.7 Power Series with Complex Terms [551] a. Introduction of Complex Terms into Power Series. Complex Representations of the Trigonometric Function, [551] b. A Glance at the General Theory of Functions of a Complex Variable, [553] APPENDIX [555] A.1 Multiplication and Division of Series [555] a. Multiplication of Absolutely Convergent Series, [555] b. Multiplication and Division of Power Series, [556] A.2 Infinite Series and Improper Integrals [557] A.3 Infinite Products [559] A.4 Series Involving Bernoulli Numbers [562] PROBLEMS [564] Chapter [8] Trigonometric Series [571] 8.1 Periodic Functions [572] a. General Remarks. Periodic Extension of a Function, [572] b. Integrals Over a Period, [573] c. Harmonic Vibrations, [574] 8.2 Superposition of Harmonic Vibrations [576] a. Harmonics. Trigonometric Polynomials, [576] b. Beats, [577] 8.3 Complex Notation [582] a. General Remarks, [582] b. Application to Alternating Currents, [583] c. Complex Notation for Trigonometrical Polynomials, [585] d. A Trigonometric Formula, [586] 8.4 Fourier Series [587] a. Fourier Coefficients, [587] b. Basic Lemma, [588] c. Proof of ∫0∞ (sin z / z) dz = п/2 , [589] d. Fourier Expansion for the Function ø (x) = x, [591] e. The Main Theorem on Fourier Expansion, [593] 8.5 Examples of Fourier Series [598] a. Preliminary Remarks, [598] b. Expansion of the Function ø (x) = x2, [598] c. Expansion of x cos x, [598] d. The Function f(x) = |x|, [600] e. A Piecewise Constant Function, [600] f. The Function sin |x|, [601] g. Expansion of cos μx. Resolution of the Cotangent into Partial Fractions. The Infinite Product for the Sine, [602] h. Further Examples, [603] 8.6 Further Discussion of Convergence [604] a. Results, [604] b. Bessel’s Inequality, [604] c. Proof of Corollaries (a), (b), and (c), [605] d. Order of Magnitude of the Fourier Coefficients Differentiation of Fourier Series, [607] 8.7 Approximation by Trigonometric and Rational Polynomials [608] a. General Remark on Representations of Functions, [608] b. Weierstrass Approximation Theorem, [608] c. Fejers Trigonometric Approximation of Fourier Polynomials by Arithmetical Means, [610] d. Approximation in the Mean and Parseval’s Relation, [612] APPENDIX I [614] A.I.l Stretching of the Period Interval. Fourier’s Integral Theorem [614] A.I.2 Gibb’s Phenomenon at Points of Discontinuity [616] A.I.3 Integration of Fourier Series [618] APPENDIX II [619] A.II.l Bernoulli Polynomials and Their Applications [619] a. Definition and Fourier Expansion, [619] b. Generating Functions and the Taylor Series of the Trigonometric and Hyperbolic Cotangent, [621] c. The Euler-Maclaurin Summation Formula, [624] d. Applications. Asymptotic Expressions, [626] e. Sums of Power Recursion Formula for Bernoulli Numbers, [628] f. Euler’s Constant and Stirling’s Series, [629] PROBLEMS [631] Chapter 9 Differential Equations for the Simplest Types of Vibration [633] 9.1 Vibration Problems of Mechanics and Physics [634] a. The Simplest Mechanical Vibrations, [634] b. Electrical Oscillations, [635] 9.2 Solution of the Homogeneous Equation. Free Oscillations [636] a. The Fomal Solution, [636] b. Physical Interpretation of the Solution, [638] c. Fulfilment of Given Initial Conditions. Uniqueness of the Solution, [639] 9.3 The Nonhomogeneous Equation. Forced Oscillations [640] a. General Remarks. Superposition, [640] b. Solution of the Nonhomogeneous Equation, [642] c. The Resonance Curve, [643] d. Further Discussion of the Oscillation, [646] e. Remarks on the Construction of Recording Instruments, [647] List of Biographical Dates [650] Index [653]
Chapter 1 Functions of Several Variables and Their Derivatives 1.1 Points and Points Sets in the Plane and in Space [1] a. Sequences of points. Convergence, [1] b. Sets of points in the plane, [3] c. The boundary of a set. Closed and open sets, [6] d. Closure as set of limit points, [9] e. Points and sets of points in space, [9] 1.2 Functions of Several Independent Variables [11] a. Functions and their domains, [11] b. The simplest types of functions, [12] c. Geometrical representation of functions, [13] 1.3 Continuity [17] a. Definition, [17] b. The concept of limit of a function of several variables, [19] c. The order to which a function vanishes, [22] 1.4 The Partial Derivatives of a Function [26] a. Definition. Geometrical representation, [26] b. Examples, [32] c. Continuity and the existence of partial derivatives, [34] d. Change of the order of differentiation, [36] 1.5 The Differential of a Function and Its Geometrical Meaning [40] a. The concept of differentiability, [40] b. Directional derivatives, [43] c. Geometric interpretation of differentiability, The tangent plane, [46] d. The total differential of a function, [49] e. Application to the calculus of errors, [52] 1.6 Functions of Functions (Compound Functions) and the Introduction of New Independent Variables [53] a. Compound functions. The chain rule, [53] b. Examples, [59] c. Change of independent variables, [60] 1.7 The Mean Value Theorem and Taylor’s Theorem for Functions of Several Variables [64] a. Preliminary remarks about approximation by polynomials, [64] b. The mean value theorem, [66] c. Taylor’s theorem for several independent variables, [68] 1.8 Integrals of a Function Depending on a Parameter [71] a. Examples and definitions, [71] b. Continuity and differentiability of an integral with respect to the parameter, [74] c. Interchange of integrations. Smoothing of functions, [80] 1.9 Differentials and Line Integrals [82] a. Linear differential forms, [82] b. Line integrals of linear differential forms, [85] c. Dependence of line integrals on endpoints, [92] 1.10 The Fundamental Theorem on Integrability of Linear Differential Forms [95] a. Integration of total differentials, [95] b. Necessary conditions for line integrals to depend only on the end points, [96] c. Insufficiency of the integrability conditions, [98] d. Simply connected sets, [102] e. The fundamental theorem, [104] APPENDIX A.l. The Principle of the Point of Accumulation in Several Dimensions and Its Applications [107] a. The principle of the point of accumulation, [107] b. Cauchy’s convergence test. Compactness, [108] c. The Heine-Borel covering theorem, [109] d. An application of the Heine-Borel theorem to closed sets contains in open sets, [110.] A.2. Basic Properties of Continuous Functions [112] A.3. Basic Notions of the Theory of Point Sets [113] a. Sets and sub-sets, [113] b. Union and intersection of sets, [115] c. Applications to sets of points in the plane, [117.] A.4. Homogeneous functions. [119] Chapter 2 Vectors, Matrices, Linear Transformations [122] 2.1 Operations with Vectors a. Definition of vectors, [122] b. Geometric representation of vectors, [124] c. Length of vectors. Angles between directions, [127] d. Scalar products of vectors, [131] e. Equation of hyperplanes in vector form, [133] f. Linear dependence of vectors and systems of linear equations, [136] 2.2 Matrices and Linear Transformations [143] a. Change of base. Linear spaces, [143] b. Matrices, [146] c. Operations with matrices, [150] d. Square matrices. The reciprocal of a matrix. Orthogonal matrices. [153] 2.3 Determinants [159] a. Determinants of second and third order, [159] b. Linear and multilinear forms of vectors, [163] c. Alternating multilinear forms. Definition of determinants, [166] d. Principal properties of determinants, [171] e. Application of determinants to systems of linear equations. [175] 2.4 Geometrical Interpretation of Determinants [180] a. Vector products and volumes of parallelepipeds in three-dimensional space, [180] b. Expansion of a determinant with respect to a column. Vector products in higher dimensions, [187] c. Areas of parallelograms and volumes of parallelepipeds in higher dimensions, [190] d. Orientation of parallelepipeds in n-dimen-sional space, [195] e. Orientation of planes and hyperplanes, [200] f. Change of volume of parallelepipeds in linear transformations, [201] 2.5 Vector Notions in Analysis [204] a. Vector fields, [204] b. Gradient of a scalar, [205] c. Divergence and curl of a vector field, [208] d. Families of vectors. Application to the theory of curves in space and to motion of particles, [211] Chapter 3 Developments and Applications of the Differential Calculus 3.1 Implicit Functions [218] a. General remarks, [218] b. Geometrical interpretation, [219] c. The implicit function theorem, [221] d. Proof of the implicit function theorem, [225] e. The implicit function theorem for more than two independent variables, [228] 3.2 Curves and Surfaces in Implicit Form [230] a. Plane curves in implicit form, [230] b. Singular points of curves, [236] c. Implicit representation of surfaces, [238] 3.3 Systems of Functions, Transformations, and Mappings [241] a. General remarks, [241] b. Curvilinear coordinates, [246] c. Extension to more than two independent variables, [249] d. Differentiation formulae for the inverse functions, [252] e. Symbolic product of mappings, [257] f. General theorem on the inversion of transformations and of systems of implicit functions. Decomposition into primitive mappings, [261] g. Alternate construction of the inverse mapping by the method of successive approximations, [266] h. Dependent functions, [268] i. Concluding remarks, [275] 3.4 Applications a. Elements of the theory of surfaces, [278] b. Conformal transformation in general, 289 [278] 3.5 Families of Curves, Families of Surfaces, and Their Envelopes [290] a. General remarks, [290] b. Envelopes of one-parameter families of curves, [292] c. Examples, [296] d. Endevelopes of families of surfaces, [303] 3.6 Alternating Differential Forms [307] a. Definition of alternating differential forms, [307] b. Sums and products of differential forms, [310] c. Exterior derivatives of differential forms, [312] d. Exterior differential forms in arbitrary coordinates, [316] 3.7 Maxima and Minima [325] a. Necessary conditions, [325] b. Examples, [327] c. Maxima and minima with subsidiary conditions, [330] d. Proof of the method of undetermined multipliers in the simplest case, [334] e. Generalization of the method of undetermined multipliers, [337] f. Examples, [340] APPENDIX A.l Sufficient Conditions for Extreme Values [345] A.2 Numbers of Critical Points Related to Indices of a Vector Field [352] A.3 Singular Points of Plane Curves [360] A.4 Singular Points of Surfaces [362] A.5 Connection Between Euler’s and Lagrange’s Representation of the motion of a Fluid [363] A.6 Tangential Representation of a Closed Curve and the Isoperimetric Inequality [365]
Chapter 4 Multiple Integrals 4.1 Areas in the Plane [367] a. Definition of the Jordan measure of area, 367 b. A set that does not have an area, [370] c. Rules for operations with areas, [372] 4.2 Double Integrals [374] a. The double integral as a volume, [374] b. The general analytic concept of the integral, [376] c. Examples, [379] d. Notation. Extensions. Fundamental rules, [381] e. Integral estimates and the mean value theorem, [383] 4.3 Integrals over Regions in three and more Dimensions [385] 4.4 Space Differentiation. Mass and Density [386] 4.5 Reduction of the Multiple Integral to Repeated Single Integrals [388] a. Integrals over a rectangle, [388] b. Change of order of integration. Differentiation under the integral sign, [390] c. Reduction of double integrals to single integrals for more general regions, [392] d. Extension of the results to regions in several dimensions, [397] 4.6 Transformation of Multiple Integrals [398] a. Transformation of integrals in the plane, [398] b. Regions of more than two dimensions, [403] 4.7 Improper Multiple Integrals [406] a. Improper integrals of functions over bounded sets, [407] b. Proof of the general convergence theorem for improper integrals, [411] c. Integrals over unbounded regions,414 4.8 Geometrical Applications [417] a. Elementary calculation of volumes, [417] b. General remarks on the calculation of volumes. Solids of revolution. Volumes in spherical coordinates, [419] c. Area of a curved surface, [421] 4.9 Physical Applications [431] a. Moments and center of mass,431 b. Moments of inertia, [433] c. The compound pendulum, [436] d. Potential of attracting masses, [438] 4.10 Multiple Integrals in Curvilinear Coordinates [445] a. Resolution of multiple integrals, [445] b. Application to areas swept out by moving curves and volumes swept out by moving surfaces. Guldin’s formula. The polar planimeter, [448] 4.11 Volumes and Surface Areas in Any Number of Dimensions [453] a. Surface areas and surface integrals in more than three dimensions, [453] b. Area and volume of the n-dimensional sphere, [455] c. Generalizations. Parametric Representations, [459] 4.12 Improper Single Integrals as Functions of a Parameter [462] a. Uniform convergence. Continuous dependence on the parameter, [462] b. Integration and differentiation of improper integrals with respect to a parameter, [466] c. Examples, [469] d. Evaluation of Fresnel’s integrals, [473] 4.13 The Fourier Integral [476] a. Introduction, [476] b. Examples, [479] c. Proof of Fourier’s integral theorem, [481] d. Rate of convergence in Fourier’s integral theorem, [485] e. Parseval’s identity for Fourier transforms, [488] f. The Fourier transformation for functions of several variables, [490] 4.14 The Eulerian Integrals (Gamma Function) [497] a. Definition and functional equation, [497] b. Convex functions. Proof of Bohr and Mollerup’s theorem, [499] c. The infinite products for the gamma function, [503] d. The nextensio theorem, [507] e. The beta function, [508] f. Differentiation and integration of fractional order. Abel’s integral equation, [511] APPENDIX: DETAILED ANALYSIS OF THE PROCESS OF INTEGRATION A.l Area [515] a. Subdivisions of the plane and the corresponding inner and outer areas, [515] b. Jordan-measurable sets and their areas, 517 c. Basic properties of areas, [519] A.2 Integrals of Functions of Several Variables [524] a. Definition of the integral of a function f(x, y), [524] b. Integrability of continuous functions and integrals over sets, [526] c. Basic rules for multiple integrals, 528 d. Reduction of multiple integrals to repeated single integrals, [531] A.3 Transformation of Areas and Integrals [534] a. Mappings of sets, [534] b. Trans formation of multiple integrals,539 A.4 Note on the Definition of the Area of a Curved Surface [540] Chapter 5 Relations Between Surface and Volume Integrals 5.1 Connection Between Line Integrals and Double Integrals in the Plane (The Integral Theorems of Gauss, Stokes, and Green) [543] 5.2 Vector Form of the Divergence Theorem. Stokes’s Theorem [551] 5.3 Formula for Integration by Parts in Two Dimensions. Green’s Theorem [556] 5.4 The Divergence Theorem Applied to the Transformation of Double Integrals [558] a. The case of 1-1 mappings, [558] b. Transformation of integrals and degree of mapping, [561] 5.5 Area Differentiation. Transformation of ∆u to Polar Coordinates [565] 5.6 Interpretation of the Formulae of Gauss and Stokes by Two-Dimensional Flows [569] 5.7 Orientation of Surfaces [575] a. Orientation of two-dimensional surfaces in three-space, [575] b. Orientation of curves on oriented surfaces, [587] 5.8 Integrals of Differential Forms and of Scalars over Surfaces [589] a. Double integrals over oriented plane regions, [589] b. Surface integrals of second-order differential forms, [592] c. Relation between integrals of differential forms over oriented surfaces to integrals of scalars over unoriented surfaces, [594] 5.9 Gauss’s and Green’s Theorems in Space [597] a. Gauss’s theorem, [597] b. Application of Gauss’s theorem to fluid flow, [602] c. Gauss’s theorem applied to space forces and surface forces, [605] d. Integration by parts and Green’s theorem in three dimensions, [607] e. Application of Green’s theorem to the transformation of AU to spherical coordinates, [608] 5.10 Stokes’s Theorem in Space [611] a. Statement and proof of the theorem, [611] b. Interpretation of Stokes’s theorem, [615] 5.11 Integral Identities in Higher Dimensions [622] APPENDIX: GENERAL THEORY OF SURFACES AND OF SURFACE INTEGALS A.l Surfaces and Surface Integrals in Three dimensions [624] a. Elementary surfaces, [624] b. Integral of a function over an elementary surface, [627] c.Oriented elementary surfaces, [629] d. Simple surfaces, [631] e. Partitions of unity and integrals over simple surfaces, [634] A.2 The Divergence Theorem [637] a. Statement of the theorem and its invariance, [637] b. Proof of the theorem, [639] A.3 Stokes’s Theorem [642] A.4 Surfaces and Surface Integrals in Euclidean Spaces of Higher Dimensions a. Elementary surfaces, [645] b. Integral of a differential form over an oriented elementary surface, [647] c. Simple m-dimensional surfaces, 648 [645] A.5 Integrals over Simple Surfaces, Gauss’s Divergence Theorem, and the General Stokes Formula in Higher Dimensions [651] Chapter 6 Differential Equations 6.1 The Differential Equations for the Motion of a Particle in Three Dimensions [654] a. The equations of motion, [654] b. The principle of conservation of energy, [656] c. Equilibrium. Stability, [659] d. Small oscillations about a position of equilibrium, [661] e. Planetary motion, [665] f. Boundary value problems. The loaded cable and the loaded beam, [672] 6.2 The General Linear Differential Equation of the First Order [678] a. Separation of variables, [678] b. The linear first-order equation, [680] 6.3 Linear Differential Equations of Higher Order [683] a. Principle of superposition. General solutions, [683] b. Homogeneous differential equations of the second second order, [688] c. The non-homogeneous differential equations. Method of variation of parameters, [691] 6.4 General Differential Equations of the First Order [697] a. Geometrical interpretation, [697] b. The differential equation of a family of curves. Singular solutions. Orthogonal trajectories, [699] c. Theorem of the existence and uniqueness of the solution, [702] 6.5 Systems of Differential Equations and Differential Equations of Higher Order [709] 6.6 Integration by the Method of Undermined Coefficients [711] 6.7 The Potential of Attracting Charges and Laplace’s Equation [713] a. Potentials of mass distributions, [713] b. The differential equation of the potential, [718] c. Uniform double layers, 719 d. The mean value theorem, [722] e. Boundary value problem for the circle. Poisson’s integral, [724] 6.8 Further Examples of Partial Differential Equations from Mathematical Physics [727] a. The wave equation in one dimension, [727] b. The wave equation in three-dimensional space, [728] c. Maxwell’s equations in free space, [731] Chapter 7 Calculus of Variations 7.1 Functions and Their Extrema [737] 7.2 Necessary conditions for Extreme Values of a Functional [741] a. Vanishing of the first variation, [741] b. Deduction of Euler’s differential equation, [743] c. Proofs of the fundamental lemmas, [747] d. Solution of Euler’s differential equation in special cases. Examples, [748] e. Identical vanishing of Euler’s expression, [752] 7.3 Generalizations [753] a. Integrals with more than one argument function, [753] b. Examples, [755] c. Hamilton’s principle. Lagrange’s equations, [757] d. Integrals involving higher derivatives, [759] e. Several independent variables, [760] 7.4 Problems Involving Subsidiary Conditions. Lagrange Multipliers [762] a. Ordinary subsidiary conditions,762 b. Other types of subsidiary conditions, [765] Chapter 8 Functions of a Complex Variable 8.1 Complex Functions Represented by Power Series [769] a. Limits and infinite series with complex terms, [769] b. Power series, [772] c. Differentiation and integration of power series, [773] d. Examples of power series, [776] 8.2 Foundations of the General Theory of Functions of a Complex Variable [778] a. The postulate of differentiability, [778] b. The simplest operations of the differential calculus, [782] c. Conformal transformation. Inverse functions, [785] 8.3 The Integration of Analytic Functions [787] a. Definition of the integral, [787] b. Cauchy’s theorem, [789] c. Applications. The logarithm, the exponential function, and the general power function, [792] 8.4 Cauchy’s Formula and Its Applications [797] a. Cauchy’s formula, [797] b. Expansion of analytic functions in power series, [799] c. The theory of functions and potential theory, [802] d. The converse of Cauchy’s theorem, [803] e. Zeros, poles, and residues of an analytic function, [803] 8.5 Applications to Complex Integration (Contour Integration) [807] a. Proof of the formula (8.22), [807] b. Proof of the formula (8.22), [808] c. Application of the theorem of residues to the integration of rational functions, [809] d. The theorem of residues and linear differential equations with constant coefficients, [812] 8.6 Many-Valued Functions and Analytic Extension [814] List of Biographical Dates [941] Index [943]
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
|---|---|---|---|---|---|---|---|---|
Libros
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 26 C858i (Browse shelf) | Vol. 1 | Available | A-2604 | ||
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Libros
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 26 C858i (Browse shelf) | Vol. 2 | Available | A-4558 |
Basado en la obra del primer autor, Vorlesungen über Differential- und Intergralrechnung, y en sus subsiguientes traducciones al inglés, publicadas por primera vez en 1934 bajo el título: Differential and integral calculus.
Vol. 2 publicado por Wiley. "A Wiley-Interscience publication."
Chapter 1 Introduction [1] --
1.1 The Continuum of Numbers [1] --
a. The System of Natural Numbers and Its Extension. Counting and Measuring, [1] --
b. Real Numbers and Nested Intervals, [7] --
c. Decimal Fractions. Bases Other Than Ten, [9] --
d. Definition of Neighborhood, [12] --
e. Inequalities, [12] --
1.2 The Concept of Function [17] --
a. Mapping-Graph, [18] --
b. Definition of the Concept of Functions of a Continuous Variable. Domain and Range of a Function, [21] --
c. Graphical Representation. Monotonic Functions, [24] --
d. Continuity, 31 e. The Intermediate Value Theorem. Inverse Functions, [44] --
1.3 The Elementary Functions [47] --
a. Rational Functions, [47] --
b. Algebraic Functions, [49] --
c. Trigonometric Functions, [49] --
d. The Exponential Function and the Logarithm, [51] --
e. Compound Functions, Symbolic Products, Inverse Functions, [52] --
1.4 Sequences [55] --
1.5 Mathematical Induction [57] --
1.6 The Limit of a Sequence [60] --
a. an = 1/n, [61] --
b. a2m = 1/m; a2m-1 = 1/2m, [62] --
c. an = n/n+1, [63] --
d. an = n√p , [64] --
e. an = αn , [65] --
f. Geometrical Illustration of the Limits of αn and n√p, [65] --
g. The Geometric Series, [67] --
h. an = n√n , [69] --
i. an = (√n+1) - √n, [69] --
1.7 Further Discussion of the Concept of Limit [70] --
a. Definition of Convergence and Divergence, [70] --
b. Rational Operations with Limits, [71] --
c. Intrinsic Convergence Tests. Monotone Sequences, [73] --
d. Infinite Series and the Summation Symbol, [75] --
e. The Number e, [77] --
f. The Number п as a Limit, [80] --
1.8 The Concept of Limit for Functions of a Continuous Variable [82] --
a. Some Remarks about the Elementary Functions, [86] --
Supplements [87] --
5.1 Limits and the Number Concept [89] --
a. The Rational Numbers, [89] --
b. Real Numbers Determined by Nested Sequences of Rational Intervals, [90] --
c. Order, Limits, and Arithmetic Operations for Real Numbers, [92] --
d. Completeness of the Number Continuum. Compactness of Closed Intervals. Convergence Criteria, [94] --
e. Least Upper Bound and Greatest Lower Bound, [97] --
f. Denumerability of the Rational Numbers, [98] --
5.2 Theorems on Continuous Functions [99] --
5.3 Polar Coordinates [101] --
S.4 Remarks on Complex Numbers [103] --
PROBLEMS [106] --
Chapter The Fundamental Ideas of the Integral and Differential Calculus [119] --
2.1 The Integra] [120] --
a. Introduction, [120] --
b. The Integral as an Area, [121] --
c. Analytic Definition of the Integral. Notations, [122] --
2.2 Elementary Examples of Integration [128] --
a. Integration of Linear Function, [128] --
b. Integration of x2, [130] --
c. Integration of xα for Integers α ≠ -1, [131] --
d. Integration of xα for Rational α Other Than -1, [134] --
e. Integration of sin x and cos x, [135] --
2.3 Fundamental Rules of Integration [136] --
a. Additivity, [136] --
b. Integral of a Sum of a Product with a Constant, [137] --
c. Estimating Integrals, 138, --
d. The Mean Value Theorem for Integrals, [139] --
2.4 The Integral as a Function of the Upper Limit (Indefinite Integral) [143] --
2.5 Logarithm Defined by an Integral [145] --
a. Definition of the Logarithm Function, [145] --
b. The Addition Theorem for Logarithms, [147] --
2.6 Exponential Function and Powers [149] --
a. The Logarithm of the Number e, [149] --
b. The Inverse Function of the Logarithm. The Exponential Function, [150] --
c. The Exponential Function as Limit of Powers, [152] --
d. Definition of Arbitrary Powers of Positive Numbers, [152] --
e. Logarithms to Any Base, [153] --
2.7 The Integral of an Arbitrary Power of x [154] --
2.8 The Derivative [155] --
a. The Derivative and the Tangent, [156] --
b. The Derivative as a Velocity, [162] --
c. Examples of Differentiation, [163] --
d. Some Fundamental Rules for Differentiation, [165] --
e. Differentiability and Continuity of Functions, [166] --
f. Higher Derivatives and Their Significance, [169] --
g. Derivative and Difference Quotient. Leibnitz’s Notation, [171] --
h. The Mean Value Theorem of Differential Calculus, [173] --
i. Proof of the Theorem, [175] --
j. The Approximation of Functions by Linear Functions. Definition of Differentials, [179] --
k. Remarks on Applications to the Natural Sciences, [183] --
2.9 The Integral, the Primitive Function, and the Fundamental Theorems of the Calculus [184] --
a. The Derivative of the Integral, [184] --
b. The Primitive Function and Its Relation to the Integral, [186] --
c. The Use of the Primitive Function for Evaluation of Definite Integrals, [189] --
d. Examples, [191] --
Supplement The Existence of the Definite Integral of a Continuous Function [192] --
PROBLEMS [196] --
Chapter 3 The Techniques of Calculus [201] --
Part A Differentiation and Integration of the Elementary Functions [201] --
3.1 The Simplest Rules for Differentiation and Their Applications [201] --
a. Rules for Differentiation, [201] --
b. Differentiation of the Rational Functions, [204] --
c. Differentiation of the Trigonometric Functions, [205] --
3.2 The Derivative of the Inverse Function [206] --
a. General Formula, [206] --
b. The Inverse of the nth Power; the nth Root, [210] --
c. The Inverse Trigonometric Functions—Multivaluedness, [210] --
d. The Corresponding Integral Formulas, [215] --
e. Derivative and Integral of the Exponential Function, [216] --
3.3 Differentiation of Composite Functions [217] --
a. Definitions, [217] --
b. The Chain Rule, [218] --
c. The Generalized Mean Value Theorem of the Differential Calculus, [222] --
3.4 Some Applications of the Exponential Function [223] --
a. Definition of the Exponential Function by Means of a Differential Equation, [223] --
b. Interest Compounded Continuously. Radioactive Disintegration, [224] --
c. Cooling or Heating of a Body by a Surrounding Medium, [225] --
d. Variation of the Atmospheric Pressure with the Height above the Surface of the Earth, [226] --
e. Progress of a Chemical Reaction, [227] --
f. Switching an Electric Circuit on or off, [228] --
3.5 The Hyperbolic Functions [228] --
a. Analytical Definition, [228] --
b. Addition Theorems and Formulas for Differentiation [231] --
c. The Inverse Hyperbolic Functions, [232] --
d. Further Analogies, [234] --
3.6 Maxima and Minima [236] --
a. Convexity and Concavity of Curves, [236] --
b. Maxima and Minima—Relative Extrema. Stationary Points, [238] --
3.7 The Order of Magnitude of Functions [248] --
a. The Concept of Order of Magnitude. The Simplest Cases, [248] --
b. The Order of Magnitude of the Exponential Function and of the Logarithm, 249 c. General Remarks, [251] --
d. The Order of Magnitude of a Function in the Neighborhood of an Arbitrary Point, [252] --
e. The Order of Magnitude (or Smallness) of a Function Tending to Zero, [252] --
f. The “O” and “o” Notation for Orders of Magnitude, [253] --
APPENDIX [255] --
A.1 Some Special Functions [255] --
a. The Function y = e1/x2 , [255] --
b. The Function y = e1/x, [256] --
c. The Function y = tanh 1/x, [257] --
d. The Function y — x tanh 1/x, [258] --
e. The Function y - x sin 1/x, y(0) = 0, [259] --
A.2 Remarks on the Differentiability of Functions [259] --
Part B Techniques of Integration [261] --
3.8 Table of Elementary Integrals [263] --
3.9 The Method of Substitution [263] --
a. The Substitution Formula. Integral of a Composite Function, [263] --
b. A Second Derivation of the Substitution Formula, [268] --
c. Examples. Integration Formulas, [270] --
3.10 Further Examples of the Substitution Method [271] --
3.11 Integration by Parts [274] --
a. General Formula, [274] --
b. Further Examples of Integration by Parts, [276] --
c. Integral Formula for (b) + f(a) [278] --
d. Recursive Formulas, [278] --
e. Wallis’s Infinite Product for п, [280] --
3.12 Integration of Rational Functions [282] --
a. The Fundamental Types, [283] --
b. Integration of the Fundamental Types, [284] --
e. Partial Fractions, [286] --
d. Examples of Resolution into Partial Fractions. Method of Undetermined Coefficients, [288] --
3.13 Integration of Some Other Classes of Functions [290] --
a. Preliminary Remarks on the Rational Representation of the Circle and the Hyperbola, [290] --
b. Integration of R(cos x, sin x), [293] --
c. Integration of R(cosh x, sinh x), [294] --
d. Integration of r(x, √(l — x2)), [294] --
e. Integration of r(x, √(x2— 1)), [295] --
f. Integration of r(x, √(ax2 + 2bx + c)), [295] --
g. Integration of R(x, Vox2 + 2bx + c), [295] --
h. Further Examples of Reduction to Integrals of Rational Functions, [296] --
i. Remarks on the Examples, [297] --
Part C Further Steps in the Theory of Integral Calculus [298] --
3.14 Integrals of Elementary Functions [298] --
a. Definition of Functions by Integrals. Elliptic Integrals and Functions, [298] --
b. On Differentiation and Integration, [300] --
3.15 Extension of the Concept of Integral [301] --
a. Introduction. Definition of “Improper” Integrals, [301] --
b. Functions with Infinite Discontinuities, [303] --
c. Interpretation as Areas, 304 d. Tests for Convergence, [305] --
e. Infinite Interval of Integration, [306] --
f. The Gamma Function, [308] --
g. The Dirichlet Integral, [309] --
h. Substitution. Fresnel Integrals, [310] --
3.16 The Differential Equations of the Trigonometric Functions [312] --
a. Introductory Remarks on Differential Equations, [312] --
b. Sin x and cos x defined by a Differential Equation and Initial Conditions, [312] --
PROBLEMS [314] --
Chapter 4 Applications in Physics and Geometry [324] --
4.1 Theory of Plane Curves [324] --
a. Parametric Representation, [324] --
b. Change of Parameters, [326] --
c. Motion along a Curve. Time as the Parameter. Example of the Cycloid, [328] --
d. Classifications of Curves. Orientation, [333] --
e. Derivatives. Tangent and Normal, in Parametric Representation, [343] --
f. The Length of a Curve, [348] --
g. The Arc Length as a Parameter, [352] --
h. Curvature, [354] --
i. Change of Coordinate Axes. Invariance, [360] --
j. Uniform Motion in the Special Theory of Relativity, [363] --
k. Integrals Expressing Area within Closed Curves, [365] --
l. Center of Mass and Moment of a Curve, [373] --
m. Area and Volume of a Surface of Revolution, [374] --
n. Moment of Inertia, [375] --
4.2 Examples [376] --
a. The Common Cycloid, [376] --
b. The Catenary, [378] --
c. The Ellipse and the Lemniscate, [378] --
4.3 Vectors in Two Dimensions [379] --
a. Definition of Vectors by Translation. Notations, [380] --
b. Addition and Multiplication of Vectors, [384] --
c. Variable Vectors, Their Derivatives, and Integrals, [392] --
d. Application to Plane Curves. Direction, Speed, and Acceleration, [394] --
4.4 Motion of a Particle under Given Forces [397] --
a. Newton’s Law of Motion, [397] --
b. Motion of Falling Bodies, [398] --
c. Motion of a Particle Constrained to a Given Curve, [400] --
4.5 Free Fall of a Body Resisted by Air [402] --
4.6 The Simplest Type of Elastic Vibration [404] --
4.7 Motion on a Given Curve [405] --
a. The Differential Equation and Its Solution, [405] --
b. Particle Sliding down a Curve, [407] --
c. Discussion of the Motion, [409] --
d. The Ordinary Pendulum, [410] --
e. The Cycloidal Pendulum, [411] --
4.8 Motion in a Gravitational Field [413] --
a. Newton’s Universal Law of Gravitation, [413] --
b. Circular Motion about the Center of Attraction, [415] --
c. Radial Motion—Escape Velocity, [416] --
4.9 Work and Energy [418] --
a. Work Done by Forces during a Motion, [418] --
b. Work and Kinetic Energy. Conservation of Energy, [420] --
c. The Mutual Attraction of Two Masses, [421] --
d. The Stretching of a Spring, [423] --
e. The Charging of a Condenser, [423] --
APPENDIX [424] --
A.l Properties of the Evolute [424] --
A.2 Areas Bounded by Closed Curves. Indices [430] --
PROBLEMS [435] --
Chapter 5 Taylor's Expansion [440] --
5.1 Introduction: Power Series [440] --
5.2 Expansion of the Logarithm and the Inverse Tangent [442] --
a. The Logarithm, 442 b. The Inverse Tangent, [444] --
5.3 Taylor’s Theorem [445] --
a. Taylor’s Representation of Polynomials, [445] --
b. Taylor’s Formula for Nonpolynomial Functions, [446] --
5.4 Expression and Estimates for the Remainder [447] --
a. Cauchy’s and Lagrange’s Expressions, [447] --
b. An Alternative Derivation of Taylor’s Formula, [450] --
5.5 Expansions of the Elementary Functions [453] --
a. The Exponential Function, [453] --
b. Expansion of sin x, cos x, sinh x, cosh x, [454] --
c. The Binomial Series, [456] --
5.6 Geometrical Applications [457] --
a. Contact of Curves, [458] --
b. On the Theory of Relative Maxima and Minima, [461] --
APPENDIX I [462] --
A.I.1 Example of a Function Which Cannot Be Expanded in a Taylor Series [462] --
A.I.2 Zeros and Infinites of Functions [463] --
a. Zeros of Order n, 463 b. Infinity of Order v, [463] --
A.I.3 Indeterminate Expressions [464] --
A.I.4 The Convergence of the Taylor Series of a Function with Nonnegative Derivatives of all Orders [467] --
APPENDIX H INTERPOLATION [470] --
A.II.1 The Problem of Interpolation. Uniqueness [470] --
A.II.2 Construction of the Solution. Newton’s Interpolation Formula [471] --
A.n.3 The Estimate of the Remainder [474] --
A.II.4 The Lagrange Interpolation Formula [476] --
PROBLEMS [477] --
Chapter 6 Numerical Methods [481] --
6.1 Computation of Integrals [482] --
a. Approximation by Rectangles, [482] --
b. Refined Approximations—Simpson’s Rule, [483] --
6.2 Other Examples of Numerical Methods [490] --
a. The “Calculus of Errors”, [490] --
b. Calculation of п, 492 c. Calculation of Logarithms, [493] --
6.3 Numerical Solution of Equations [494] --
a. Newton’s Method, 495 b. The Rule of False Position, 497 c. The Method of Iteration, [499] --
d. Iterations and Newton’s Procedure, [502] --
APPENDIX [504] --
A.l Stirling’s Formula [504] --
PROBLEMS [507] --
Chapter 7 Infinite Sums and Products [510] --
7.1 The Concepts of Convergence and Divergence [511] --
a. Basic Concepts, [511] --
b. Absolute Convergence and Conditional Convergence, [513] --
c. Rearrangement of Terms, [517] --
d. Operations with Infinite Series, [520] --
7.2 Tests for Absolute Convergence and Divergence [520] --
a. The Comparison Test. Majorants, [520] --
b. Convergence Tested by Comparison with the Geometric Series, [521] --
c. Comparison with an Integral, [524] --
7.3 Sequences of Functions [526] --
a. Limiting Processes with Functions and Curves, [527] --
7.4 Uniform and Nonuniform Convergence [529] --
a. General Remarks and Definitions, [529] --
b. A Test for Uniform Convergence, [534] --
c. Continuity of the Sum of a Uniformly Convergent Series of Continuous Functions, [535] --
d. Integration of Uniformly Convergent Series, [536] --
e. Differentiation of Infinite Series, [538] --
7.5 Power Series [540] --
a. Convergence Properties of Power Series— Interval of Convergence, [540] --
b. Integration and Differentiation of Power Series, [542] --
c. Operations with Power Series, [543] --
d. Uniqueness of Expansion, [544] --
e. Analytic Functions, [545] --
7.6 Expansion of Given Functions in Power Series. Method of Undetermined Coefficients. Examples [546] --
a. The Exponential Function, [546] --
b. The Binomial Series, [546] --
c. The Series for arc sin x, [549] --
d. The Series for ar sinh x = log [x + √(1 + x2)], [549] --
e. Example of Multiplication of Series, [550] --
f. Example of Term-by-Term Integration (Elliptic Integral), [550] --
7.7 Power Series with Complex Terms [551] --
a. Introduction of Complex Terms into Power Series. Complex Representations of the Trigonometric Function, [551] --
b. A Glance at the General Theory of Functions of a Complex Variable, [553] --
APPENDIX [555] --
A.1 Multiplication and Division of Series [555] --
a. Multiplication of Absolutely Convergent Series, [555] --
b. Multiplication and Division of Power Series, [556] --
A.2 Infinite Series and Improper Integrals [557] --
A.3 Infinite Products [559] --
A.4 Series Involving Bernoulli Numbers [562] --
PROBLEMS [564] --
Chapter [8] --
Trigonometric Series [571] --
8.1 Periodic Functions [572] --
a. General Remarks. Periodic Extension of a Function, [572] --
b. Integrals Over a Period, [573] --
c. Harmonic Vibrations, [574] --
8.2 Superposition of Harmonic Vibrations [576] --
a. Harmonics. Trigonometric Polynomials, [576] --
b. Beats, [577] --
8.3 Complex Notation [582] --
a. General Remarks, [582] --
b. Application to Alternating Currents, [583] --
c. Complex Notation for Trigonometrical Polynomials, [585] --
d. A Trigonometric Formula, [586] --
8.4 Fourier Series [587] --
a. Fourier Coefficients, [587] --
b. Basic Lemma, [588] --
c. Proof of ∫0∞ (sin z / z) dz = п/2 , [589] --
d. Fourier Expansion for the Function ø (x) = x, [591] --
e. The Main Theorem on Fourier Expansion, [593] --
8.5 Examples of Fourier Series [598] --
a. Preliminary Remarks, [598] --
b. Expansion of the Function ø (x) = x2, [598] --
c. Expansion of x cos x, [598] --
d. The Function f(x) = |x|, [600] --
e. A Piecewise Constant Function, [600] --
f. The Function sin |x|, [601] --
g. Expansion of cos μx. Resolution of the Cotangent into Partial Fractions. The Infinite Product for the Sine, [602] --
h. Further Examples, [603] --
8.6 Further Discussion of Convergence [604] --
a. Results, [604] --
b. Bessel’s Inequality, [604] --
c. Proof of Corollaries (a), (b), and (c), [605] --
d. Order of Magnitude of the Fourier Coefficients Differentiation of Fourier Series, [607] --
8.7 Approximation by Trigonometric and Rational Polynomials [608] --
a. General Remark on Representations of Functions, [608] --
b. Weierstrass Approximation Theorem, [608] --
c. Fejers Trigonometric Approximation of Fourier Polynomials by Arithmetical Means, [610] --
d. Approximation in the Mean and Parseval’s Relation, [612] --
APPENDIX I [614] --
A.I.l Stretching of the Period Interval. Fourier’s Integral Theorem [614] --
A.I.2 Gibb’s Phenomenon at Points of Discontinuity [616] --
A.I.3 Integration of Fourier Series [618] --
APPENDIX II [619] --
A.II.l Bernoulli Polynomials and Their Applications [619] --
a. Definition and Fourier Expansion, [619] --
b. Generating Functions and the Taylor Series of the Trigonometric and Hyperbolic Cotangent, [621] --
c. The Euler-Maclaurin Summation Formula, [624] --
d. Applications. Asymptotic Expressions, [626] --
e. Sums of Power Recursion Formula for Bernoulli Numbers, [628] --
f. Euler’s Constant and Stirling’s Series, [629] --
PROBLEMS [631] --
Chapter 9 Differential Equations for the Simplest Types of Vibration [633] --
9.1 Vibration Problems of Mechanics and Physics [634] --
a. The Simplest Mechanical Vibrations, [634] --
b. Electrical Oscillations, [635] --
9.2 Solution of the Homogeneous Equation. Free Oscillations [636] --
a. The Fomal Solution, [636] --
b. Physical Interpretation of the Solution, [638] --
c. Fulfilment of Given Initial Conditions. Uniqueness of the Solution, [639] --
9.3 The Nonhomogeneous Equation. Forced Oscillations [640] --
a. General Remarks. Superposition, [640] --
b. Solution of the Nonhomogeneous Equation, [642] --
c. The Resonance Curve, [643] --
d. Further Discussion of the Oscillation, [646] --
e. Remarks on the Construction of Recording Instruments, [647] --
List of Biographical Dates [650] --
Index [653] --
Chapter 1 Functions of Several Variables and Their Derivatives --
1.1 Points and Points Sets in the Plane and in Space [1] --
a. Sequences of points. Convergence, [1] --
b. Sets of points in the plane, [3] --
c. The boundary of a set. Closed and open sets, [6] --
d. Closure as set of limit points, [9] --
e. Points and sets of points in space, [9] --
1.2 Functions of Several Independent Variables [11] --
a. Functions and their domains, [11] --
b. The simplest types of functions, [12] --
c. Geometrical representation of functions, [13] --
1.3 Continuity [17] --
a. Definition, [17] --
b. The concept of limit of a function of several variables, [19] --
c. The order to which a function vanishes, [22] --
1.4 The Partial Derivatives of a Function [26] --
a. Definition. Geometrical representation, [26] --
b. Examples, [32] --
c. Continuity and the existence of partial derivatives, [34] --
d. Change of the order of differentiation, [36] --
1.5 The Differential of a Function and Its Geometrical Meaning [40] --
a. The concept of differentiability, [40] --
b. Directional derivatives, [43] --
c. Geometric interpretation of differentiability, The tangent plane, [46] --
d. The total differential of a function, [49] --
e. Application to the calculus of errors, [52] --
1.6 Functions of Functions (Compound Functions) and the Introduction of New Independent Variables [53] --
a. Compound functions. The chain rule, [53] --
b. Examples, [59] --
c. Change of independent variables, [60] --
1.7 The Mean Value Theorem and Taylor’s Theorem for Functions of Several Variables [64] --
a. Preliminary remarks about approximation by polynomials, [64] --
b. The mean value theorem, [66] --
c. Taylor’s theorem for several independent variables, [68] --
1.8 Integrals of a Function Depending on a Parameter [71] --
a. Examples and definitions, [71] --
b. Continuity and differentiability of an integral with respect to the parameter, [74] --
c. Interchange of integrations. Smoothing of functions, [80] --
1.9 Differentials and Line Integrals [82] --
a. Linear differential forms, [82] --
b. Line integrals of linear differential forms, [85] --
c. Dependence of line integrals on endpoints, [92] --
1.10 The Fundamental Theorem on Integrability of Linear Differential Forms [95] --
a. Integration of total differentials, [95] --
b. Necessary conditions for line integrals to depend only on the end points, [96] --
c. Insufficiency of the integrability conditions, [98] --
d. Simply connected sets, [102] --
e. The fundamental theorem, [104] --
APPENDIX --
A.l. The Principle of the Point of Accumulation in Several Dimensions and Its Applications [107] --
a. The principle of the point of accumulation, [107] --
b. Cauchy’s convergence test. Compactness, [108] --
c. The Heine-Borel covering theorem, [109] --
d. An application of the Heine-Borel theorem to closed sets contains in open sets, [110.] --
A.2. Basic Properties of Continuous Functions [112] --
A.3. Basic Notions of the Theory of Point Sets [113] --
a. Sets and sub-sets, [113] --
b. Union and intersection of sets, [115] --
c. Applications to sets of points in the plane, [117.] --
A.4. Homogeneous functions. [119] --
Chapter 2 Vectors, Matrices, Linear Transformations [122] --
2.1 Operations with Vectors --
a. Definition of vectors, [122] --
b. Geometric representation of vectors, [124] --
c. Length of vectors. Angles between directions, [127] --
d. Scalar products of vectors, [131] --
e. Equation of hyperplanes in vector form, [133] --
f. Linear dependence of vectors and systems of linear equations, [136] --
2.2 Matrices and Linear Transformations [143] --
a. Change of base. Linear spaces, [143] --
b. Matrices, [146] --
c. Operations with matrices, [150] --
d. Square matrices. The reciprocal of a matrix. Orthogonal matrices. [153] --
2.3 Determinants [159] --
a. Determinants of second and third order, [159] --
b. Linear and multilinear forms of vectors, [163] --
c. Alternating multilinear forms. Definition of determinants, [166] --
d. Principal properties of determinants, [171] --
e. Application of determinants to systems of linear equations. [175] --
2.4 Geometrical Interpretation of Determinants [180] --
a. Vector products and volumes of parallelepipeds in three-dimensional space, [180] --
b. Expansion of a determinant with respect to a column. Vector products in higher dimensions, [187] --
c. Areas of parallelograms and volumes of parallelepipeds in higher dimensions, [190] --
d. Orientation of parallelepipeds in n-dimen-sional space, [195] --
e. Orientation of planes and hyperplanes, [200] --
f. Change of volume of parallelepipeds in linear transformations, [201] --
2.5 Vector Notions in Analysis [204] --
a. Vector fields, [204] --
b. Gradient of a scalar, [205] --
c. Divergence and curl of a vector field, [208] --
d. Families of vectors. Application to the theory of curves in space and to motion of particles, [211] --
Chapter 3 Developments and Applications of the Differential Calculus --
3.1 Implicit Functions [218] --
a. General remarks, [218] --
b. Geometrical interpretation, [219] --
c. The implicit function theorem, [221] --
d. Proof of the implicit function theorem, [225] --
e. The implicit function theorem for more than two independent variables, [228] --
3.2 Curves and Surfaces in Implicit Form [230] --
a. Plane curves in implicit form, [230] --
b. Singular points of curves, [236] --
c. Implicit representation of surfaces, [238] --
3.3 Systems of Functions, Transformations, and Mappings [241] --
a. General remarks, [241] --
b. Curvilinear coordinates, [246] --
c. Extension to more than two independent variables, [249] --
d. Differentiation formulae for the inverse functions, [252] --
e. Symbolic product of mappings, [257] --
f. General theorem on the inversion of transformations and of systems of implicit functions. Decomposition into primitive mappings, [261] --
g. Alternate construction of the inverse mapping by the method of successive approximations, [266] --
h. Dependent functions, [268] --
i. Concluding remarks, [275] --
3.4 Applications --
a. Elements of the theory of surfaces, [278] --
b. Conformal transformation in general, 289 [278] --
3.5 Families of Curves, Families of Surfaces, and Their Envelopes [290] --
a. General remarks, [290] --
b. Envelopes of one-parameter families of curves, [292] --
c. Examples, [296] --
d. Endevelopes of families of surfaces, [303] --
3.6 Alternating Differential Forms [307] --
a. Definition of alternating differential forms, [307] --
b. Sums and products of differential forms, [310] --
c. Exterior derivatives of differential forms, [312] --
d. Exterior differential forms in arbitrary coordinates, [316] --
3.7 Maxima and Minima [325] --
a. Necessary conditions, [325] --
b. Examples, [327] --
c. Maxima and minima with subsidiary conditions, [330] --
d. Proof of the method of undetermined multipliers in the simplest case, [334] --
e. Generalization of the method of undetermined multipliers, [337] --
f. Examples, [340] --
APPENDIX --
A.l Sufficient Conditions for Extreme Values [345] --
A.2 Numbers of Critical Points Related to Indices of a Vector Field [352] --
A.3 Singular Points of Plane Curves [360] --
A.4 Singular Points of Surfaces [362] --
A.5 Connection Between Euler’s and Lagrange’s Representation of the motion of a Fluid [363] --
A.6 Tangential Representation of a Closed Curve and the Isoperimetric Inequality [365] --
Chapter 4 Multiple Integrals --
4.1 Areas in the Plane [367] --
a. Definition of the Jordan measure of area, 367 b. A set that does not have an area, [370] --
c. Rules for operations with areas, [372] --
4.2 Double Integrals [374] --
a. The double integral as a volume, [374] --
b. The general analytic concept of the integral, [376] --
c. Examples, [379] --
d. Notation. Extensions. Fundamental rules, [381] --
e. Integral estimates and the mean value theorem, [383] --
4.3 Integrals over Regions in three and more Dimensions [385] --
4.4 Space Differentiation. Mass and Density [386] --
4.5 Reduction of the Multiple Integral to Repeated Single Integrals [388] --
a. Integrals over a rectangle, [388] --
b. Change of order of integration. Differentiation under the integral sign, [390] --
c. Reduction of double integrals to single integrals for more general regions, [392] --
d. Extension of the results to regions in several dimensions, [397] --
4.6 Transformation of Multiple Integrals [398] --
a. Transformation of integrals in the plane, [398] --
b. Regions of more than two dimensions, [403] --
4.7 Improper Multiple Integrals [406] --
a. Improper integrals of functions over bounded sets, [407] --
b. Proof of the general convergence theorem for improper integrals, [411] --
c. Integrals over unbounded regions,414 --
4.8 Geometrical Applications [417] --
a. Elementary calculation of volumes, [417] --
b. General remarks on the calculation of volumes. Solids of revolution. Volumes in spherical coordinates, [419] --
c. Area of a curved surface, [421] --
4.9 Physical Applications [431] --
a. Moments and center of mass,431 --
b. Moments of inertia, [433] --
c. The compound pendulum, [436] --
d. Potential of attracting masses, [438] --
4.10 Multiple Integrals in Curvilinear Coordinates [445] --
a. Resolution of multiple integrals, [445] --
b. Application to areas swept out by moving curves and volumes swept out by moving surfaces. Guldin’s formula. The polar planimeter, [448] --
4.11 Volumes and Surface Areas in Any Number of Dimensions [453] --
a. Surface areas and surface integrals in more than three dimensions, [453] --
b. Area and volume of the n-dimensional sphere, [455] --
c. Generalizations. Parametric Representations, [459] --
4.12 Improper Single Integrals as Functions of a Parameter [462] --
a. Uniform convergence. Continuous dependence on the parameter, [462] --
b. Integration and differentiation of improper integrals with respect to a parameter, [466] --
c. Examples, [469] --
d. Evaluation of Fresnel’s integrals, [473] --
4.13 The Fourier Integral [476] --
a. Introduction, [476] --
b. Examples, [479] --
c. Proof of Fourier’s integral theorem, [481] --
d. Rate of convergence in Fourier’s integral theorem, [485] --
e. Parseval’s identity for Fourier transforms, [488] --
f. The Fourier transformation for functions of several variables, [490] --
4.14 The Eulerian Integrals (Gamma Function) [497] --
a. Definition and functional equation, [497] --
b. Convex functions. Proof of Bohr and Mollerup’s theorem, [499] --
c. The infinite products for the gamma function, [503] --
d. The nextensio theorem, [507] --
e. The beta function, [508] --
f. Differentiation and integration of fractional order. Abel’s integral equation, [511] --
APPENDIX: DETAILED ANALYSIS OF THE PROCESS OF INTEGRATION --
A.l Area [515] --
a. Subdivisions of the plane and the corresponding inner and outer areas, [515] --
b. Jordan-measurable sets and their areas, 517 c. Basic properties of areas, [519] --
A.2 Integrals of Functions of Several Variables [524] --
a. Definition of the integral of a function f(x, y), [524] --
b. Integrability of continuous functions and integrals over sets, [526] --
c. Basic rules for multiple integrals, 528 d. Reduction of multiple integrals to repeated single integrals, [531] --
A.3 Transformation of Areas and Integrals [534] --
a. Mappings of sets, [534] --
b. Trans formation of multiple integrals,539 --
A.4 Note on the Definition of the Area of a Curved Surface [540] --
Chapter 5 Relations Between Surface and Volume Integrals --
5.1 Connection Between Line Integrals and Double Integrals in the Plane (The Integral Theorems of Gauss, Stokes, and Green) [543] --
5.2 Vector Form of the Divergence Theorem. Stokes’s Theorem [551] --
5.3 Formula for Integration by Parts in Two Dimensions. Green’s Theorem [556] --
5.4 The Divergence Theorem Applied to the Transformation of Double Integrals [558] --
a. The case of 1-1 mappings, [558] --
b. Transformation of integrals and degree of mapping, [561] --
5.5 Area Differentiation. Transformation of ∆u to Polar Coordinates [565] --
5.6 Interpretation of the Formulae of Gauss and Stokes by Two-Dimensional Flows [569] --
5.7 Orientation of Surfaces [575] --
a. Orientation of two-dimensional surfaces in three-space, [575] --
b. Orientation of curves on oriented surfaces, [587] --
5.8 Integrals of Differential Forms and of Scalars over Surfaces [589] --
a. Double integrals over oriented plane regions, [589] --
b. Surface integrals of second-order differential forms, [592] --
c. Relation between integrals of differential forms over oriented surfaces to integrals of scalars over unoriented surfaces, [594] --
5.9 Gauss’s and Green’s Theorems in Space [597] --
a. Gauss’s theorem, [597] --
b. Application of Gauss’s theorem to fluid flow, [602] --
c. Gauss’s theorem applied to space forces and surface forces, [605] --
d. Integration by parts and Green’s theorem in three dimensions, [607] --
e. Application of Green’s theorem to the transformation of AU to spherical coordinates, [608] --
5.10 Stokes’s Theorem in Space [611] --
a. Statement and proof of the theorem, [611] --
b. Interpretation of Stokes’s theorem, [615] --
5.11 Integral Identities in Higher Dimensions [622] --
APPENDIX: GENERAL THEORY OF SURFACES AND OF SURFACE --
INTEGALS --
A.l Surfaces and Surface Integrals in Three dimensions [624] --
a. Elementary surfaces, [624] --
b. Integral of a function over an elementary surface, [627] --
c.Oriented elementary surfaces, [629] --
d. Simple surfaces, [631] --
e. Partitions of unity and integrals over simple surfaces, [634] --
A.2 The Divergence Theorem [637] --
a. Statement of the theorem and its invariance, [637] --
b. Proof of the theorem, [639] --
A.3 Stokes’s Theorem [642] --
A.4 Surfaces and Surface Integrals in Euclidean Spaces of Higher Dimensions --
a. Elementary surfaces, [645] --
b. Integral of a differential form over an oriented elementary surface, [647] --
c. Simple m-dimensional surfaces, 648 [645] --
A.5 Integrals over Simple Surfaces, Gauss’s Divergence Theorem, and the General Stokes Formula in Higher Dimensions [651] --
Chapter 6 Differential Equations --
6.1 The Differential Equations for the Motion of a Particle in Three Dimensions [654] --
a. The equations of motion, [654] --
b. The principle of conservation of energy, [656] --
c. Equilibrium. Stability, [659] --
d. Small oscillations about a position of equilibrium, [661] --
e. Planetary motion, [665] --
f. Boundary value problems. The loaded cable and the loaded beam, [672] --
6.2 The General Linear Differential Equation of the First Order [678] --
a. Separation of variables, [678] --
b. The linear first-order equation, [680] --
6.3 Linear Differential Equations of Higher Order [683] --
a. Principle of superposition. General solutions, [683] --
b. Homogeneous differential equations of the second second order, [688] --
c. The non-homogeneous differential equations. Method of variation of parameters, [691] --
6.4 General Differential Equations of the First Order [697] --
a. Geometrical interpretation, [697] --
b. The differential equation of a family of curves. Singular solutions. Orthogonal trajectories, [699] --
c. Theorem of the existence and uniqueness of the solution, [702] --
6.5 Systems of Differential Equations and Differential Equations of Higher Order [709] --
6.6 Integration by the Method of Undermined Coefficients [711] --
6.7 The Potential of Attracting Charges and Laplace’s Equation [713] --
a. Potentials of mass distributions, [713] --
b. The differential equation of the potential, [718] --
c. Uniform double layers, 719 d. The mean value theorem, [722] --
e. Boundary value problem for the circle. Poisson’s integral, [724] --
6.8 Further Examples of Partial Differential Equations from Mathematical Physics [727] --
a. The wave equation in one dimension, [727] --
b. The wave equation in three-dimensional space, [728] --
c. Maxwell’s equations in free space, [731] --
Chapter 7 Calculus of Variations --
7.1 Functions and Their Extrema [737] --
7.2 Necessary conditions for Extreme Values of a Functional [741] --
a. Vanishing of the first variation, [741] --
b. Deduction of Euler’s differential equation, [743] --
c. Proofs of the fundamental lemmas, [747] --
d. Solution of Euler’s differential equation in special cases. Examples, [748] --
e. Identical vanishing of Euler’s expression, [752] --
7.3 Generalizations [753] --
a. Integrals with more than one argument function, [753] --
b. Examples, [755] --
c. Hamilton’s principle. Lagrange’s equations, [757] --
d. Integrals involving higher derivatives, [759] --
e. Several independent variables, [760] --
7.4 Problems Involving Subsidiary Conditions. Lagrange Multipliers [762] --
a. Ordinary subsidiary conditions,762 --
b. Other types of subsidiary conditions, [765] --
Chapter 8 Functions of a Complex Variable --
8.1 Complex Functions Represented by Power Series [769] --
a. Limits and infinite series with complex terms, [769] --
b. Power series, [772] --
c. Differentiation and integration of power series, [773] --
d. Examples of power series, [776] --
8.2 Foundations of the General Theory of Functions of a Complex Variable [778] --
a. The postulate of differentiability, [778] --
b. The simplest operations of the differential calculus, [782] --
c. Conformal transformation. Inverse functions, [785] --
8.3 The Integration of Analytic Functions [787] --
a. Definition of the integral, [787] --
b. Cauchy’s theorem, [789] --
c. Applications. The logarithm, the exponential function, and the general power function, [792] --
8.4 Cauchy’s Formula and Its Applications [797] --
a. Cauchy’s formula, [797] --
b. Expansion of analytic functions in power series, [799] --
c. The theory of functions and potential theory, [802] --
d. The converse of Cauchy’s theorem, [803] --
e. Zeros, poles, and residues of an analytic function, [803] --
8.5 Applications to Complex Integration (Contour Integration) [807] --
a. Proof of the formula (8.22), [807] --
b. Proof of the formula (8.22), [808] --
c. Application of the theorem of residues to the integration of rational functions, [809] --
d. The theorem of residues and linear differential equations with constant coefficients, [812] --
8.6 Many-Valued Functions and Analytic Extension [814] --
List of Biographical Dates [941] --
Index [943] --
MR, 90j:00002a
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