In this post, we will see the book The Remarkable Sine Functions by A. I. Markushevich.

About the book

In the present book, we shall show how it is possible, by beginning with other curves (such as the equilateral hyperbola or Bernoulli’s lemniscate (a curve having the form of a figure- eight), to define interesting and important functions analogous to the trigonometric functions, similar to them in some respects but possessing certain new characteristics. These functions are called respectively hyperbolic and lemniscate functions. In analogy with them, we shall refer to the trigonometric functions as circular functions.

The reader is assumed to have a familiarity with the ele­ ments of analytic geometry and differential and integral calculus. The necessary material on integration in the complex plane will be given in the present book though proofs will be omitted.

The ultimate purpose of the book is to acquaint the reader not possessing an extensive knowledge of the theory of functions of a complex variable with the simplest representatives of the class of elliptic functions, namely, lemniscate functions and the somewhat more general Jacobian elliptic functions.

In conclusion, we warn the reader that this book is not in­ tended for light reading. He must read it with his pencil in his hand.

The book was translated from Russian by Scripta Technica and was published in 1966.

Credits to the original uploader.

You can get the book here.

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Contents

Preface

1. Geometric Definition of Circular, Hyperbolic And Lemniscate Functions 1

2. Generalized Sines 13

3. Integration in the Complex Plane 25

4. Euler’s Method of Deriving the Addition Theorems 41

5. Further Study of Complex Values of the Argument 49

6. Zeros and Poles. Simple and Double Periodicity. The Concept of an Elliptic Function 73

Index 99