INTRODUCTORY MATHEMATICAL ANALYSIS
CONTENTS EUCLIDEAN SPACE VECTORS TOPOLOGY OF RN LIMITS OF FUNCTIONS SEQUENCES AND SERIES OF FUNCTIONS SEQUENCES AND SERIES OF REAL OR COMPLEX NUMBERS SEQUENCES OF FUNCTIONS UNIFORM CONVERGENCE SERIES OF FUNCTIONS POWER SERIES: RADIUS OF CONVERGENCE TAYLOR–MACLAURIN SERIES BINOMIAL SERIES THE ERROR FUNCTION ABEL THEOREM AND SERIES SUMMATION BASEL PROBLEM EXTENSION OF ELEMENTARY FUNCTIONS TO THE COMPLEX ?ELD COMPLEX EXPONENTIAL COMPLEX GONIOMETRIC HYPERBOLIC FUNCTIONS COMPLEX LOGARITHM EXERCISES SOLVED EXERCISES UNSOLVED EXERCISES MULTIDIMENSIONAL DI?ERENTIAL CALCULUS PARTIAL DERIVATIVES DI?ERENTIABILITY MAXIMA AND MINIMA SU?CIENT CONDITIONS LAGRANGE MULTIPLIERS MEAN–VALUE THEOREM IMPLICIT FUNCTION THEOREM ORDINARY DI?ERENTIAL EQUATIONS OF ?RST ORDER: GENERAL THEORY PRELIMINARY NOTIONS SYSTEMS OF ODES: EQUATIONS OF HIGH ORDER EXISTENCE OF SOLUTIONS: PEANO THEOREM EXISTENCE AND UNIQUENESS: PICARD–LINDEL¨OHF THEOREM INTERVAL OF EXISTENCE VECTOR–VALUED DI?ERENTIAL EQUATIONS SOLUTION CONTINUATION ORDINARY DI?ERENTIAL EQUATIONS OF ?RST ORDER: METHODS FOR EXPLICIT SOLUTIONS SEPARABLE EQUATIONS SINGULAR INTEGRALS HOMOGENEOUS EQUATIONS QUASI HOMOGENEOUS EQUATIONS EXACT EQUATIONS INTEGRATING FACTOR FOR NON–EXACT EQUATIONS LINEAR EQUATIONS OF ?RST ORDER BERNOULLI EQUATION RICCATI EQUATION CROSS–RATIO PROPERTY REDUCED FORM OF THE RICCATI EQUATION CONNECTION WITH THE LINEAR EQUATION OF SECOND ORDER 510 CHANGE OF VARIABLE 5101 EXERCISES LINEAR DI?ERENTIAL EQUATIONS OF SECOND ORDER OPERATOR NOTATION WRONSKIAN DETERMINANT ORDER REDUCTION CONSTANT–COE?CIENT EQUATIONS CAUCHY–EULER EQUATIONS INVARIANT AND NORMAL FORM NON–HOMOGENEOUS EQUATION VARIATION OF PARAMETERS NON–HOMOGENEOUS EQUATIONS WITH CONSTANT COE?CIENTS PROLOGUE TO MEASURE THEORY SET THEORY SETS INDEXES AND CARTESIAN PRODUCT CARTESIAN PRODUCT FUNCTIONS EQUIVALENCES REAL INTERVALS CARDINALITY THE REAL NUMBER SYSTEM THE EXTENDED REAL NUMBER SYSTEM TOPOLOGY CLOSED SETS LIMIT CLOSURE COMPACTNESS CONTINUITY LEBESGUE INTEGRAL MEASURE THEORY ?–ALGEBRAS BOREL SETS MEASURES TRANSLATION INVARIANCE SIMPLE FUNCTIONS INTEGRAL OF SIMPLE FUNCTIONS MEASURABLE FUNCTIONS ALMOST EVERYWHERE CONNECTION WITH RIEMANN INTEGRAL THE RIEMANN INTEGRAL LEBESGUE–VITALI THEOREM AN INTERESTING EXAMPLE NON LEBESGUE INTEGRALS DIRAC MEASURE DISCRETE MEASURE GENERATION OF MEASURES 810 PASSAGE TO THE LIMIT 8101 MONOTONE CONVERGENCE 81011 ANALYTIC FUNCTIONS 8102 EXERCISES 8103 DOMINATED CONVERGENCE 8104 EXERCISES 8105 A PROPERTY OF INCREASING FUNCTIONS 811 DI?ERENTIATION UNDER THE INTEGRAL SIGN 8111 THE PROBABILITY INTEGRAL (1) 8112 THE PROBABILITY INTEGRAL (2) 8113 EXERCISES 812 BASEL PROBLEM AGAIN 813 DEBYE INTEGRAL RADON–NIKODYM THEOREM SIGNED MEASURES 10 MULTIPLE INTEGRALS 101 INTEGRATION IN R2 1011 SMART APPLICATIONS OF FUBINI THEOREM 10111 FRESNEL INTEGRALS 10112 CAUCHY FORMULA FOR ITERATED INTEGRATION 10113 A CHALLENGING DE?NITE INTEGRAL 10114 FRULLANI INTEGRAL 10115 BASEL PROBLEM, ONCE MORE 102 CHANGE OF VARIABLE 103 INTEGRATION IN RN 1031 EXERCISES 104 PRODUCT OF ?–ALGEBRAS 11 GAMMA AND BETA FUNCTIONS 111 GAMMA FUNCTION 112 BETA FUNCTION 1121 ? AND THE PROBABILITY INTEGRAL 1122 LEGENDRE DUPLICATION FORMULA 1123 EULER RE?EXION FORMULA 113 DE?NITE INTEGRALS 114 DOUBLE INTEGRATION TECHNIQUES 12 FOURIER TRANSFORM ON THE REAL LINE 121 FOURIER TRANSFORM 1211 EXAMPLES 122 PROPERTIES OF THE FOURIER TRANSFORM 1221 LINEARITY 1222 THE SHIFT THEOREM 1223 THE STRETCH THEOREM 1224 COMBINING SHIFTS AND STRETCHES 123 CONVOLUTION 124 LINEAR ORDINARY DI?ERENTIAL EQUATIONS 125 EXERCISES 13 PARABOLIC EQUATIONS 131 PARTIAL DI?ERENTIAL EQUATIONS 1311 CLASSI?CATION OF SECOND–ORDER LINEAR PARTIAL DI?ERENTIAL EQUATIONS 132 THE HEAT EQUATION 1321 UNIQUENESS OF SOLUTION: HOMOGENEOUS CASE 1322 FUNDAMENTAL SOLUTIONS: HEAT KERNEL 1323 INITIAL DATA ON (0, ?) 133 PARABOLIC EQUATIONS WITH CONSTANT COE?CIENTS 1331 EXERCISES 134 BLACK–SCHOLES EQUATION 1341 EXERCISES 135 NON–HOMOGENEOUS EQUATION: DUHAMEL INTEGRAL BIBLIOGRAPHY