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The Integration of Functions, of a Single Variable (Classic Reprint)
by:G. H. Hardy
Introduction The problem considered in the following pages is what is sometimes called the problem of indefinite integration or of finding a function whose diflferential coefficient is a given function .T hese descriptions are vague and in some ways misleading; and it is necessary to define our problem more precisely before we proceed further. Let us...
Introduction The problem considered in the following pages is what is sometimes called the problem of indefinite integration or of finding a function whose diflferential coefficient is a given function .T hese descriptions are vague and in some ways misleading; and it is necessary to define our problem more precisely before we proceed further. Let us suppose for the moment that f(x) is a real continuous function of the real variable x. We wish to determine a function y, whose differential coefficient is (), or to solve the equation !=(-) (1)- A little reflection shows that this problem may be analysed into a number of parts. We wish, first, to know whether such a function as ynecessarily exists, whether the equation (l) has always a solution; whether the solution, if it exists, is unique; and what relations hold between different solutions, if there are more than one. The answers to these questions are contained in that part of the theory of functions of a real variable which deals with definite integrals .T he definite integral y=r(t)dt (2), Ja which is defined as the limit of a certain sum, is a solution of the equation (1). Further ?+G (3), where Gis an arbitrary constant, is also a solution, and all solutions of (1) are of the form (3). H.(Typographical errors above are due to OCR software and don't occur in the book.)About the Publisher Forgotten Books is a publisher of historical writings, such as: Philosophy, Classics, Science, Religion, History, Folklore and Mythology.Forgotten Books' Classic Reprint Series utilizes the latest technology to regenerate facsimiles of historically important writings. Careful attention has been made to accurately preserve the original format of each page whilst digitally enhancing the aged text. Read books online for free at www.forgottenbooks.org
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