Elliptic Integrals by Harris Hancock (PDF)

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The editors of the present series of mathematical monographs have requested me to write a work on elliptic integrals which shall relate almost entirely to the three well-known elliptic integrals, with tables and examples showing practical applications, and which shall fill about one hundred octavo pages. In complying with their request, I shall limit the monograph to what is known as theL egendre-J acobi theory; and to keep the work within the desired number of pages I must confine the discussion almost entirely to what is known as the elliptic integrals of the first and second kinds. In the elementary calculus are found methods of integrating any rational expression involving under a square root sign a quadratic in one variable; in the present work, which may be regarded as a somewhat more advanced calculus; we have to integrate similar expressions where cubics and quarries in one variable occur under the root sign. Whatever be the nature of these cubics and quarries, it will be seen that the integrals may be transformed into standard normal forms. Tables are given of these normal forms, so that the integral in question may be calculated to any degree of exactness required. With the trigonometric sine function is associated its inverse function, an integral; and similarly with the normal forms of elliptic integrals there are associated elliptic functions. A short account is given of these functions which emphasizes their doubly periodic properties. By making suitable transformations and using the inverse of these functions, it is found that the integrals in question may be expressed more concisely through the normal forms and in a manner that indicates the transformation employed.(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, Folklo

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⭐Harris Hancock is the author of a three volume treatise on elliptic functions. This is not that. Rather this book is a short introduction to elliptic integrals and functions which makes an excellent book to read early on in one’s studies of these topics.The approach is historical beginning with Legendre’s work on integration of certain algebraic functions leading to the classification of the three kinds of elliptic integrals. Primary treatment is given to the first two kinds as they are the simplest. Chapter one has nice graphs of these integral functions.Chapter two moves to Jacobi’s work with the introduction of the am function along with sn, cn, and dn. It also covers Jacobi’s imaginary transformation and provides a bunch more graphs and figures which really help make the material more concrete.The third chapter is a long, hard slog through pages of equations showing how various integrals can be reduced to Legendre’s forms from chapter 1. Many important results are to be found here.The fourth chapter concerns itself with numerical computations of elliptic integrals and Landen’s transformation. This chapter does develop some interesting expansions for the elliptic integrals, but is of limited use unless you still use a slide rule. Also, I found the treatment of Jacobi’s construction quite opaque.Chapter five is a very short chapter of additional examples and some problems.Chapter six ends the book with a collection of five place tables for the hand calculating crowd.Quite an excellent book and littered with good problems and figures, my only real complaint was the limited scope. In particular, you will learn nothing of Weirstrass’ treatment of elliptic functions from this work. I’d highly recommend this book, but I’d also suggest reading

⭐either before or in conjunction with this work.

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