Mathematical Analysis: Approximation and Discrete Processes

Contents......Page 00008.djvu
Preface......Page 00006.djvu
1.1 Introduction......Page 00014.djvu
a. Numbers and measurement......Page 00015.djvu
b. Never-ending processes......Page 00017.djvu
c. Back to numbers......Page 00019.djvu
d. An axiomatic or a constructive approach?......Page 00021.djvu
1.2.1 Algebraic and order properties......Page 00022.djvu
b. Axioms for multiplication......Page 00023.djvu
c. The distributive law......Page 00024.djvu
d. Order......Page 00025.djvu
a. Supremum......Page 00026.djvu
c. Dedekind cuts of R......Page 00028.djvu
1.2.3 Uniqueness of reals......Page 00029.djvu
a. Natural numbers and the principle of induction......Page 00030.djvu
b. Approximation of reals by rational numbers......Page 00033.djvu
c. Recursive statements......Page 00034.djvu
1.4 Summing Up......Page 00038.djvu
1.5 Exercises......Page 00039.djvu
2.1 Sequences......Page 00044.djvu
a. Limit of a sequence......Page 00048.djvu
b. Properties of limits and calculus......Page 00049.djvu
c. Limits of monotone sequences......Page 00052.djvu
e. Subsequences......Page 00053.djvu
a. The principle of nested intervals or Cantor\'s principle......Page 00054.djvu
b. Cauchy criterion......Page 00055.djvu
c. Upper and lower limits......Page 00057.djvu
e. The continuity property of the reals......Page 00059.djvu
a. Limits of sequences and limits of functions......Page 00060.djvu
b. Continuity in terms of sequences......Page 00061.djvu
2.4 Some Special Sequences......Page 00062.djvu
a. Elementary limits......Page 00063.djvu
b. Powers, exponentials and factorials......Page 00066.djvu
c. Wallis and Stirling formulas......Page 00068.djvu
d. Numerical integration......Page 00070.djvu
a. A definition of a^x using continuity......Page 00072.djvu
c. Derivative of the exponential......Page 00075.djvu
2.6 Summing Up......Page 00076.djvu
2.7 Exercises......Page 00078.djvu
3.1.1 Euclid\'s algorithm......Page 00084.djvu
a. The greatest common divisor......Page 00085.djvu
b. Integer solutions of first order equations......Page 00087.djvu
3.1.2 Prime factorization......Page 00090.djvu
3.1.3 Linear congruences......Page 00092.djvu
3.1.4 Euler\'s function \\Phi......Page 00095.djvu
3.1.5 RSA Cryptography......Page 00097.djvu
3.2 Combinatorics......Page 00101.djvu
a. Ordered samples and mappings......Page 00102.djvu
b. Nonordered samples and subsets......Page 00104.djvu
c. Ordered lists......Page 00105.djvu
d. The formula of inclusion and exclusion......Page 00106.djvu
e. Surjective maps......Page 00107.djvu
3.2.3 Location problems......Page 00108.djvu
3.2.4 The hypergeometric and multinomial distributions......Page 00110.djvu
3.3.1 The mathematical analysis of infinity......Page 00112.djvu
a. Cardinality......Page 00113.djvu
b. Cantor-Bernstein theorem......Page 00115.djvu
c. Denumerable sets......Page 00116.djvu
d. The axiom of choice......Page 00117.djvu
e. The power of the continuum......Page 00118.djvu
f. The continuum hypothesis......Page 00119.djvu
3.3.2 Some information on the theory of sets......Page 00120.djvu
3.4 Summing Up......Page 00124.djvu
3.5 Exercises......Page 00126.djvu
4. Complex Numbers......Page 00134.djvu
a. The system of complex numbers......Page 00135.djvu
b. The n-th roots......Page 00141.djvu
c. Complex exponential and logarithm......Page 00142.djvu
a. Definitions......Page 00144.djvu
b. Weierstrass\'s theorem......Page 00145.djvu
4.3.1 A few applications of the complex notation......Page 00146.djvu
4.3.2 A few applicatons to elementary Euclidean geometry......Page 00148.djvu
a. Special points of a triangle......Page 00149.djvu
b. Equilateral triangles......Page 00151.djvu
4.4 Summing Up......Page 00153.djvu
4.5 Exercises......Page 00155.djvu
5.1 Polynomials......Page 00158.djvu
5.1.1 The Division Algorithm......Page 00160.djvu
a. Euclid\'s algorithm and Bezout identity......Page 00161.djvu
c. The factor theorem......Page 00163.djvu
a. Factorization in C......Page 00166.djvu
b. Simple and multiple roots of a polynomial......Page 00169.djvu
c. Factorization in R......Page 00170.djvu
5.2.1 Solutions by radicals......Page 00171.djvu
5.2.2 Distribution of the roots of a polynomial......Page 00176.djvu
b. Sturm\'s theorem......Page 00177.djvu
a. Decomposition in C......Page 00179.djvu
b. Decomposition in R......Page 00183.djvu
c. Integration of rational functions......Page 00184.djvu
5.4 Sinusoidal Functions and Their Sums......Page 00186.djvu
a. Periodic functions......Page 00187.djvu
b. Trigonometric polynomials......Page 00188.djvu
c. Spectrum and energy identity......Page 00189.djvu
d. Sampling......Page 00191.djvu
5.4.2 Sums of sinusoidal functions......Page 00194.djvu
5.5 Summing Up......Page 00196.djvu
5.6 Exercises......Page 00198.djvu
6. Series......Page 00200.djvu
6.1 Basic Facts......Page 00201.djvu
a. Definitions and examples......Page 00202.djvu
c. Series and improper integrals......Page 00205.djvu
d. Decimals......Page 00206.djvu
6.2 Taylor Series, e and \\pi......Page 00208.djvu
a. The number \\pi......Page 00211.djvu
b. More on the number e......Page 00215.djvu
6.3 Series of Nonnegative Terms......Page 00217.djvu
a. Series of positive decreasing terms......Page 00219.djvu
b. The root and ratio tests......Page 00222.djvu
c. Viète\'s formula for \\pi......Page 00224.djvu
d. Euler and Wallis formulas......Page 00225.djvu
a. Absolute convergence......Page 00227.djvu
b. Series of complex terms......Page 00228.djvu
a. Alternating series......Page 00229.djvu
c. Sequences of bounded total variation......Page 00232.djvu
d. Dirichlet and Abel theorems......Page 00234.djvu
6.6 Products of Series......Page 00235.djvu
6.7 Rearrangements......Page 00238.djvu
6.8 Summing Up......Page 00240.djvu
6.9 Exercises......Page 00243.djvu
7. Power Series......Page 00248.djvu
7.1.1 Circle of convergence......Page 00251.djvu
a. The disc and the domain of convergence......Page 00253.djvu
a. Uniform Convergence......Page 00254.djvu
b. Continuity of uniform limits......Page 00255.djvu
c. Uniform convergence of power series......Page 00256.djvu
a. Series of derivatives and of integrals......Page 00257.djvu
b. Real power series......Page 00258.djvu
c. Power series and Taylor series......Page 00260.djvu
d. Complex series......Page 00261.djvu
7.2.1 Boundary values......Page 00263.djvu
7.2.2 Product and composition of power series......Page 00266.djvu
7.2.3 Taylor series: examples......Page 00267.djvu
7.3 Some Applications......Page 00270.djvu
7.3.1 Complex functions......Page 00271.djvu
7.3.2 An alternate definition of \\pi, e and of elementary functions......Page 00273.djvu
7.3.3 Series solutions of differential equations......Page 00275.djvu
a. Generating functions......Page 00277.djvu
b. Enumerators......Page 00279.djvu
c. Exponential enumerators......Page 00281.djvu
d. A few location problems......Page 00282.djvu
e. Partitions of a set......Page 00284.djvu
a. Bernoulli numbers......Page 00287.djvu
b. Bernoulli polynomials......Page 00289.djvu
c. Euler-MacLaurin formula and Stirling\'s approximation......Page 00291.djvu
a. Definition and characterizations......Page 00293.djvu
b. Functional relations......Page 00295.djvu
c. Asymptotics of \\Gamma and \\psi......Page 00299.djvu
7.5 Summing Up......Page 00301.djvu
7.6 Exercises......Page 00302.djvu
8. Discrete Processes......Page 00310.djvu
a. First order linear difference equations......Page 00315.djvu
b. Second order homogeneous difference equations......Page 00317.djvu
c. Second order nonhomogeneous difference equations......Page 00318.djvu
d. Z-transform and Laplace transform......Page 00319.djvu
e. Fibonacci\'s numbers......Page 00321.djvu
a. Simple examples......Page 00324.djvu
b. Evaluating algorithm performance......Page 00325.djvu
c. Rate of convergence......Page 00327.djvu
a. Definitions and elementary properties......Page 00329.djvu
b. Developments as continuous fractions......Page 00334.djvu
c. Infinite continued fractions......Page 00336.djvu
d. Irrationals and approximations by rationals......Page 00339.djvu
e. Order of approximation and transcendental numbers......Page 00342.djvu
8.2 One-Dimensional Dynamical Systems......Page 00344.djvu
a. Euler\'s method......Page 00345.djvu
b. Runge-Kutta method......Page 00347.djvu
c. Models......Page 00348.djvu
8.2.2 Examples of one-dimensional dynamics......Page 00350.djvu
b. Contractive dynamics: fixed points......Page 00351.djvu
c. Sinks and sources......Page 00352.djvu
d. Periodic orbits......Page 00353.djvu
e. Periodic-doubling cascade transition to chaos......Page 00355.djvu
f. The intermittency phenomenon......Page 00357.djvu
g. Ergodic dynamics......Page 00358.djvu
8.2.3 Chaotic dynamics......Page 00361.djvu
a. Sensitive dependence on initial conditions and the Lyapunov exponent......Page 00362.djvu
c. Bernoulli\'s shift......Page 00364.djvu
d. The triangular map......Page 00366.djvu
8.2.4 Chaotic attractors, basins of attraction......Page 00367.djvu
a. Measure and dimension......Page 00370.djvu
b. Cantor sets......Page 00373.djvu
c. Iterated function systems......Page 00375.djvu
d. Dimension of the invariant set......Page 00376.djvu
8.3.1 Game of life......Page 00382.djvu
8.3.2 Fractal boundaries......Page 00383.djvu
a. Julia sets......Page 00384.djvu
8.3.3 Fractals on the computer......Page 00385.djvu
8.4 Exercises......Page 00386.djvu
A. Mathematicians and Other Scientists......Page 00390.djvu
B. Bibliographical Notes......Page 00392.djvu
C. Index......Page 00394.djvu