A Course of Mathematical Analysis, Volume I

Description

The translation of this Course was undertaken at the suggestion of Professor W. F. Osgood, whose review of the original appeared in the July number of the Bulletin of the American Mathematical Society in 1903. The lack of standard texts on mathematical subjects in the English language is too well known to require insistence. I earnestly hope that this book will help to fill the need so generally felt throughout the American mathematical world. It may be used conveniently in our system of instruction as a text for a second course in calculus, and as a book of reference it will be found valuable to an American student throughout his work. Few alterations have been made from the French text. Slight changes of notation have been introduced occasionally for convenience, and several changes and .additions have been made at the suggestion of Professor Goursat, who has very kindly interested himself in the work of translation. To him is due all the additional matter not to be found in the French text, except the footnotes which are signed, and even these, though not of his initiative, were always edited by him. I take this opportunity to express my gratitude to the author for the permission to translate the work and for the sympathetic attitude which he has consistently assumed. I am also indebted to Professor Osgood for counsel as the work progressed and for aid in doubtful matters pertaining to the translation. The publishers, Messrs. Ginn & Company, have spared no pains to make the typography excellent. Their spirit has been far from commercial in the whole enterprise, and it is their hope, as it is mine, that the publication of this book will contribute to the advance of mathematics in America. E R HENDRICK CONTENTS CHAPTER PAGE I. DERIVATIVES AND DIFFERENTIALS 1 I. Functions of a Single Variable 1 II. Functions of Several Variables 11 III. The Differential Notation 19 II. IMPLICIT FUNCTIONS. FUNCTIONAL DETERMINANTS. CHANGE OF VARIABLE 35 I. Implicit Functions ........ 35 II. Functional Determinants ...... 52 III. Transformations ... .... 61 III. TAYLOR'S SERIES. ELEMENTARY APPLICATIONS. MAXIMA AND MINIMA ........ 89 I. Taylor's Series with a Remainder. Taylor's Series . 89 II. Singular Points. Maxima and Minima . . . .110 IV. DEFINITE INTEGRALS ........ 134 I. Special Methods of Quadrature . . . . .134 II. Definite Integrals. Allied Geometrical Concepts . . 140 III. Change of Variable. Integration by Parts . . .166 IV. Generalizations of the Idea of an Integral. Improper Integrals. Line Integrals ...... 175 V. Functions defined by Definite Integrals .... 192 VI. Approximate Evaluation of Definite Integrals . .196 V. INDEFINITE INTEGRALS 208 I. Integration of Rational Functions ..... 208 II. Elliptic and Hyperelliptic Integrals .... 226 III. Integration of Transcendental Functions . . .236 VI. DOUBLE INTEGRALS ........ 250 I. Double Integrals. Methods of Evaluation. Green's Theorem 250 II. Change of Variables. Area of a Surface . . . 264 III. Generalizations of Double Integrals. Improper Integrals. Surface Integrals ....... 277 IV. Analytical and Geometrical Applications . . . 284 viii CONTENTS CHAPTER PAGE VII. MULTIPLE INTEGRALS. INTEGRATION OF TOTAL DIFFER ENTIALS 296 I. Multiple Integrals. Change of Variables . . . 296 II. Integration of Total Differentials . . . . .313 VIII. INFINITE SERIES . •. 327 I. Series of Real Constant Terms. General Properties. Tests for Convergence 327 II. Series of Complex Terms. Multiple Series . . . 350 III. Series of Variable Terms. Uniform Convergence . . 360 IX. POWER SERIES. TRIGONOMETRIC SERIES .... 375 I. Power Series of a Single Variable . . . . . 375 II. Power Series in Several Variables ..... S94 III. Implicit Functions. Analytic Curves and Surfaces . 399 IV. Trigonometric Series. Miscellaneous Series . . .411 X. PLANE CURVES 426 I. Envelopes 426 II. Curvature 433 III. Contact of Plane Curves 443 XI. SKEW CURVES 453 I. Osculating Plane ........ 453 II. Envelopes of Surfaces . . • . . . . . 459 III. Curvature and Torsion of Skew Curves .... 468 IV. Contact between Skew Curves. Contact between Curves and Surfaces ........ 486 XII. SURFACES 497 I. Curvature of Curves drawn on a Surface . . . 497 II. Asymptotic Lines. Conjugate Lines .... 506 III. Lines of Curvature . . . . . . . .514 IV. Families of Straight Lines 526 INDEX . 541