The applications of elliptic functions G Greenhill 9781177164542 Books
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The applications of elliptic functions G Greenhill 9781177164542 Books
If you are a physics or engineering student looking to sharpen your sword with elliptic functions, look no further. This is your book.Rather than approaching elliptic functions via the historical path of Legendre's elliptic integrals, then defining Jacobi's elliptic functions in terms of their inverse, and then finally showing that these latter entities can be applied to actual physical problems such as the circular pendulum, this book introduces Jacobi's elliptic functions directly as the solutions to the equations of motion of the circular pendulum.
This is great for students of physics and engineering as they can immediately apply their physical intuition to these functions. It is decidedly less good for students of mathematics.
Also in the first chapter we get a detailed treatment of the cycloidal pendulum with a demonstration of its isochrony. We also get some series expansions and tables for computation, and even the solution of the free rigid body problem as well as a surprising amount of other material.
Again, that's all in chapter one.
Chapter two takes up the topic of elliptic integrals of the first kind. It treats them quite exhaustively. Weristrass' approach is also introduced in this chapter along with some of Klein's work. There are pages and pages of formulas here.
Chapter three turns back to applications. The examples provided are many and varied. Some are quite exotic. Others are more mundane.
At the end of this chapter you will be less than 120 pages into this book, but if you even make it this far, you will have done well. However, this book is still just getting warmed up.
Chapter four covers addition theorems. Many of them. Then it applies them to the "Poristic Polygons of Poncelet" with respect to two circles. Poncelet's poristic polygons are the subject of a deep and highly inobvious result in the plane geometry of conic sections. They were widely known and studied in the late nineteenth and early twentieth centuries. But now projective geometry has fallen on hard times, and very few people have even heard of them. Nonetheless, elliptic functions are used in the context of the good old circular pendulum together with the addition theorems to make the existence of these polygons for two circles quite clear. In doing so we are given an excellent account of Jacobi's construction. This important construction, easily understood in terms of the motion of the circular pendulum, comes up over and over again in books on elliptic functions. This chapter also provides some applications of elliptic functions to spherical trigonometry.
Chapter five is a deeper, rather abstract look at addition formulas, and even goes so far as to cover Abel's Theorem.
Chapter six brings us relatively back to earth in its coverage of elliptic integrals of the second and third kind. The third kind is particularly challenging. Some of the figures in this chapter are almost laughably busy.
Chapter seven provides even more material on elliptic integrals as well as some fairly exotic applications.
Chapter eight covers the double periodicity of elliptic functions in considerable depth, but I found it surprising to encounter this quite basic aspect of these functions so late in this book.
Chapter nine is dedicated to various series and product expansions of elliptic functions, while chapter 10 finishes up the book with transformation theory. Basically, this is just finding formulas analogous to double and half angle formulas from trig and then generalizing them. In this connection, this chapter also provides quite good introductory coverage of modular functions.
This is followed by an appendix that provides a few more examples and some final problems.
And that, finally, is the end.
Like I said, if you are really looking for expertise with elliptic functions - both Jacobian and Weirstrassian, you have come to the right place. This book is extremely detailed but makes good use of the student's presumed physical background in its explanations and examples. It also demonstrates some high levels of mathematical technique, and is laden with examples, figures, and exercises. It is probably worth pointing out though that it provides absolutely no material on theta functions.
This is an excellent book even now, though many of the applications are unsurprisingly dated. However, I definitely would not recommend this book as your first read on the topic. Make sure you have at least The elementary properties of the elliptic functions, with examples and Elliptic Integrals... under your belt before you tackle this work or you are almost certain to be overwhelmed.
Good luck with it.
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The applications of elliptic functions G Greenhill 9781177164542 Books Reviews
If you are a physics or engineering student looking to sharpen your sword with elliptic functions, look no further. This is your book.
Rather than approaching elliptic functions via the historical path of Legendre's elliptic integrals, then defining Jacobi's elliptic functions in terms of their inverse, and then finally showing that these latter entities can be applied to actual physical problems such as the circular pendulum, this book introduces Jacobi's elliptic functions directly as the solutions to the equations of motion of the circular pendulum.
This is great for students of physics and engineering as they can immediately apply their physical intuition to these functions. It is decidedly less good for students of mathematics.
Also in the first chapter we get a detailed treatment of the cycloidal pendulum with a demonstration of its isochrony. We also get some series expansions and tables for computation, and even the solution of the free rigid body problem as well as a surprising amount of other material.
Again, that's all in chapter one.
Chapter two takes up the topic of elliptic integrals of the first kind. It treats them quite exhaustively. Weristrass' approach is also introduced in this chapter along with some of Klein's work. There are pages and pages of formulas here.
Chapter three turns back to applications. The examples provided are many and varied. Some are quite exotic. Others are more mundane.
At the end of this chapter you will be less than 120 pages into this book, but if you even make it this far, you will have done well. However, this book is still just getting warmed up.
Chapter four covers addition theorems. Many of them. Then it applies them to the "Poristic Polygons of Poncelet" with respect to two circles. Poncelet's poristic polygons are the subject of a deep and highly inobvious result in the plane geometry of conic sections. They were widely known and studied in the late nineteenth and early twentieth centuries. But now projective geometry has fallen on hard times, and very few people have even heard of them. Nonetheless, elliptic functions are used in the context of the good old circular pendulum together with the addition theorems to make the existence of these polygons for two circles quite clear. In doing so we are given an excellent account of Jacobi's construction. This important construction, easily understood in terms of the motion of the circular pendulum, comes up over and over again in books on elliptic functions. This chapter also provides some applications of elliptic functions to spherical trigonometry.
Chapter five is a deeper, rather abstract look at addition formulas, and even goes so far as to cover Abel's Theorem.
Chapter six brings us relatively back to earth in its coverage of elliptic integrals of the second and third kind. The third kind is particularly challenging. Some of the figures in this chapter are almost laughably busy.
Chapter seven provides even more material on elliptic integrals as well as some fairly exotic applications.
Chapter eight covers the double periodicity of elliptic functions in considerable depth, but I found it surprising to encounter this quite basic aspect of these functions so late in this book.
Chapter nine is dedicated to various series and product expansions of elliptic functions, while chapter 10 finishes up the book with transformation theory. Basically, this is just finding formulas analogous to double and half angle formulas from trig and then generalizing them. In this connection, this chapter also provides quite good introductory coverage of modular functions.
This is followed by an appendix that provides a few more examples and some final problems.
And that, finally, is the end.
Like I said, if you are really looking for expertise with elliptic functions - both Jacobian and Weirstrassian, you have come to the right place. This book is extremely detailed but makes good use of the student's presumed physical background in its explanations and examples. It also demonstrates some high levels of mathematical technique, and is laden with examples, figures, and exercises. It is probably worth pointing out though that it provides absolutely no material on theta functions.
This is an excellent book even now, though many of the applications are unsurprisingly dated. However, I definitely would not recommend this book as your first read on the topic. Make sure you have at least The elementary properties of the elliptic functions, with examples and Elliptic Integrals... under your belt before you tackle this work or you are almost certain to be overwhelmed.
Good luck with it.
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