ELEMENTS OF THE DIFFERENTIAL AND INTEGRAL CALCULUS

By V. GAMAM

CONTENTS 1 – DI?ERENTIAL AND INTEGRAL CALCULUS § 1 THE RIEMANN INTEGRAL 1 – UPPER AND LOWER INTEGRALS OF A BOUNDED FUNCTION 2 – ELEMENTARY PROPERTIES OF INTEGRALS 3 – RIEMANN SUMS THE INTEGRAL NOTATION 4 – UNIFORM LIMITS OF INTEGRABLE FUNCTIONS 5 – APPLICATION TO FOURIER SERIES AND TO POWER SERIES § 2 INTEGRABILITY CONDITIONS 6 – THE BOREL-LEBESGUE THEOREM 7 – INTEGRABILITY OF REGULATED OR CONTINUOUS FUNCTIONS 8 – UNIFORM CONTINUITY AND ITS CONSEQUENCES 9 – DI?ERENTIATION AND INTEGRATION UNDER THE SIGN 10 – SEMICONTINUOUS FUNCTIONS 11 – INTEGRATION OF SEMICONTINUOUS FUNCTIONS § 3 THE “FUNDAMENTAL THEOREM” (FT) 12 – THE FUNDAMENTAL THEOREM OF THE DI?ERENTIAL AND INTEGRAL CALCULUS 13 – EXTENSION OF THE FUNDAMENTAL THEOREM TO REGULATED FUNC- TIONS 14 – CONVEX FUNCTIONS; H¨OLDER AND MINKOWSKI INEQUALITIES § 4 INTEGRATION BY PARTS 15 – INTEGRATION BY PARTS 16 – THE SQUARE WAVE FOURIER SERIES 17 – WALLIS’ FORMULA § 5 TAYLOR’S FORMULA 18 – TAYLOR’S FORMULA § 6 THE CHANGE OF VARIABLE FORMULA 19 – CHANGE OF VARIABLE IN AN INTEGRAL 20 – INTEGRATION OF RATIONAL FRACTIONS § 7 GENERALISED RIEMANN INTEGRALS 21 – CONVERGENT INTEGRALS: EXAMPLES AND DE?NITIONS 22 – ABSOLUTELY CONVERGENT INTEGRALS 23 – PASSAGE TO THE LIMIT UNDER THE SIGN 24 – SERIES AND INTEGRALS 25 – DI?ERENTIATION UNDER THE SIGN 26 – INTEGRATION UNDER THE SIGN § 8 APPROXIMATION THEOREMS 27 – HOW TO MAKE C?A FUNCTION WHICH IS NOT 28 – APPROXIMATION BY POLYNOMIALS 29 – FUNCTIONS HAVING GIVEN DERIVATIVES AT A POINT § 9 RADON MEASURES IN R OR C 30 – RADON MEASURES ON A COMPACT SET 31 – MEASURES ON A LOCALLY COMPACT SET 32 – THE STIELTJES CONSTRUCTION 33 – APPLICATION TO DOUBLE INTEGRALS § 10 SCHWARTZ DISTRIBUTIONS 34 – DE?NITION AND EXAMPLES 35 – DERIVATIVES OF A DISTRIBUTION