Catálogo de la biblioteca Antonio Monteiro - INMABB

Reproducción de dos textos publicados originalmente en China en 1936.

CONTENTS --
Chapter I --
Convergence of Infinite Series --
1. Definitions [1] --
2. Comparison Tests for Series of Positive Terms [3] --
3. A General Test-Ratio Test [7] --
4. Cauchy’s Test by Integration [9] --
5. Tests in Case lim un+1/un =1 [13] --
6. Kummer’s Criterion [16] --
7. Continuation. Discussion [18] --
8. Alternating Series [20] --
9. Series with Positive and Negative Terms at Pleasure [22] --
10. Infinite Products [26] --
11. The Hypergeometric Series [30] --
Chapter II --
The Number System --
1. The Problem [34] --
2. Fractions [36] --
3. Negative Numbers [40] --
4. Irrational Numbers [46] --
5. The Theorem of Continuity [48] --
6. Convergence. Limits. The Fundamental Theorem [49] --
7. Addition of Irrationals [50] --
8. Limit of the Sum of Two Variables [52] --
9. Multiplication of Irrationals [53] --
10. Roots. Inequalities [57] --
11. Retrospect [58] --
Chapter III --
Point Sets. Limits. Continuity --
1. Definitions [62] --
2. An Example [65] --
3. Functions [67] --
4. Continuation [68] --
5. Limits [70] --
6. Bounded Functions [74] --
7. Three Theorems on Limits [76] --
8. Continuous Functions [80] --
9. Three Theorems on Continuous Functions [83] --
10. Uniform Continuity [87] --
11. The Covering Theorem [91] --
12. The Axiom of Choice [91] --
Chapter IV --
Derivatives. Integrals. Implicit Functions --
1. Derivatives [97] --
2. Continuous Function without a Derivative [100] --
3. Rolle’s Theorem [101] --
4. Law of the Mean [102] --
5. Differentiation of Composite Functions. Differentials [103] --
6. Taylor’s Theorem with a Remainder [105] --
7. Functions of Several Variables [105] --
8. Integral of a Continuous Function [110] --
9. Implicit Functions [117] --
10. The Existence Theorem [119] --
11. Simultaneous Systems of Equations [123] --
12. The Inverse of a Transformation [127] --
13. Identical Vanishing of the Jacobian [128] --
14. Solutions in the Large [130] --
Chapter V --
Uniform Convergence --
1. Series of Functions [132] --
2. Uniform Convergence [134] --
4. Weierstrass’s M-Test [136] --
5. Continuity [138] --
6. Power Series [141] --
6. Abel’s Lemma [146] --
7. The Binomial Series [147] --
8. Integration of Series [154] --
9. Differentiation of Series [159] --
10. Double Limits and the s (n, m) -Theorem [164] --
11. Application: Differentiation of Series [170] --
12. Condensation of Singularities [172] --
Chapter VI --
The Elementary Functions --
1. The Trigonometric Functions [176] --
2. The Logarithmic Function [183] --
3. The Exponential Function [186] --
4. A Simpler Analytic Treatment [189] --
5. Partial Fractions. Development of cot x [190] --
6. Infinite Products [196] --
Chapter VII --
Algebraic Transformations of Infinite Series --
1. Elementary Theorems [197] --
2. The Commutative Law [198] --
3. The Associative Law [199] --
4. Double Series [200] --
5. Series of Series [205] --
6. Power Series [207] --
7. Bernoulli’s Numbers [212] --
8. The Development of cot x [214] --
9. Analytic Functions of Several Variables [215] --
10. Regular Curves. Jordan Curves [218] --
Chapter VIII --
Fourier’s Series --
1. Fourier’s Series [220] --
2. Bessel’s Inequality. Normal Functions [223] --
5. Appraisal of the Fourier’s Coefficients [226] --
4. Identical Vanishing [227] --
5. The Formulas of Summation [230] --
6. Abel’s Theorem [232] --
7. Proof of Convergence [234] --
8. Continuation. The Discontinuous Case [234] --
9. The Gibbs Effect [236] --
10. Integration and Differentiation of the Expansion [240] --
11. Divergent Series [242] --
12. Summable Fourier’s Series [247] --
15. Concluding Remarks [251] --
Chapter IX --
Definite Integrals. Line Integrals --
1. Proper Integrals. Continuity [258] --
2. Continuation [261] --
3. Differentiation. Leibniz’s Rule [263] --
4. Variable Limits of Integration [266] --
5. Iterated Integral with Constant Limits [268] --
6. Proof that ϑ2u/ϑx ϑy = ϑ2u/ϑy ϑx [269] --
7. Improper Integrals [270] --
8. Double Limits [272] --
9. Uniform Convergence [272] --
10. The de la Vallee-Poussin μ (x) - Test [273] --
11. Continuity [275] --
12. Integration. Reversal of the Order [277] --
13. Leibniz’s Rule [279] --
14. Applications [281] --
15. The Gamma Function [283] --
16. Improper Integrals over a Finite Interval [284] --
17. The Beta-Function [286] --
18. Both Limits Infinite [287] --
19. Application. The B-Function in Terms of the Γ-Function [290] --
20. Rectangular Region of Integration [292] --
21. Appraisal of an Alternating Integral [294] --
22. Computation of ∫0∞ (sin x / x)dx [296] --
Applications [296] --
Duhamel’s Theorem [300] --
Line Integrals [303] --
Continuation. The Integral: ∫c Pdx + Qdy [305] --
Chapter X --
The Gamma Function --
Definition [312] --
The Difference Equation [314] --
Gauss’s Product [314] --
Agreement of the Two Definitions [319] --
Stirling’s Formula [320] --
Chapter XI --
Fourier’s Integral --
Fourier’s Integral. Heuristic Treatment [327] --
A Lemma [329] --
Continuation. The General Case [332] --
Convergence of Fourier’s Integral. Differentiation [336] --
Derived Integrals [338] --
Fourier’s Integral for Functions of Several Variables [341] --
Chapter XII --
Differential Equations. Existence Theorems The Problem [348] --
The Existence Theorem [350] --
Continuation; n Equations [357] --
The Semi-Linear Case [362] --
Dependence on Parameters [367] --
Implicit Integral Relations [371] --
Linear Differential Equations [375] --
Differential Equations of Higher Order; General Case [376] --
Complex Variables [377] --
Linear Partial Differential Equations of the First Order [378] --
The General Case [383] --
12. Change of Variables [385] --
13. The General Partial Differential Equation of the First Order; n = 2 [387] --
14. Continuation. Integration by Means of Characteristics [393] --
15. The Case of n Variables [398] --

MR, 20 #936