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With the continued advance of computing power and accessibility,
the view that "real mathematicians don't compute" no longer has any
traction for a newer generation of mathematicians. The goal in this
book is to present a coherent variety of accessible examples of
modern mathematics where intelligent computing plays a significant
role and in so doing to highlight some of the key algorithms and to
teach some of the key experimental approaches.
The problem of evaluating integrals is well known to every student
who has had a year of calculus. It was an especially important
subject in 19th century analysis and it has now been revived with
the appearance of symbolic languages. In this book, the authors use
the problem of exact evaluation of definite integrals as a starting
point for exploring many areas of mathematics. The questions
discussed in this book, first published in 2004, are as old as
calculus itself. In presenting the combination of methods required
for the evaluation of most integrals, the authors take the most
interesting, rather than the shortest, path to the results. Along
the way, they illuminate connections with many subjects, including
analysis, number theory, algebra and combinatorics. This will be a
guided tour of exciting discovery for undergraduates and their
teachers in mathematics, computer science, physics, and
engineering.
The problem of evaluating integrals is well known to every student
who has had a year of calculus. It was an especially important
subject in 19th century analysis and it has now been revived with
the appearance of symbolic languages. In this book, the authors use
the problem of exact evaluation of definite integrals as a starting
point for exploring many areas of mathematics. The questions
discussed in this book, first published in 2004, are as old as
calculus itself. In presenting the combination of methods required
for the evaluation of most integrals, the authors take the most
interesting, rather than the shortest, path to the results. Along
the way, they illuminate connections with many subjects, including
analysis, number theory, algebra and combinatorics. This will be a
guided tour of exciting discovery for undergraduates and their
teachers in mathematics, computer science, physics, and
engineering.
The subject of elliptic curves is one of the jewels of
nineteenth-century mathematics, whose masters were Abel, Gauss,
Jacobi, and Legendre. This book presents an introductory account of
the subject in the style of the original discoverers, with
references to and comments about more recent and modern
developments. It combines three of the fundamental themes of
mathematics: complex function theory, geometry, and arithmetic.
After an informal preparatory chapter, the book follows a
historical path, beginning with the work of Abel and Gauss on
elliptic integrals and elliptic functions. This is followed by
chapters on theta functions, modular groups and modular functions,
the quintic, the imaginary quadratic field, and on elliptic curves.
The many exercises with hints scattered throughout the text give
the reader a glimpse of further developments. Requiring only a
first acquaintance with complex function theory, this book is an
ideal introduction to the subject for graduate students and
researchers in mathematics and physics.
The subject of elliptic curves is one of the jewels of nineteenth-century mathematics, whose masters were Abel, Gauss, Jacobi, and Legendre. This book presents an introductory account of the subject in the style of the original discoverers, with references to and comments about more recent and modern developments. It combines three of the fundamental themes of mathematics: complex function theory, geometry, and arithmetic. After an informal preparatory chapter, the book follows a historical path, beginning with the work of Abel and Gauss on elliptic integrals and elliptic functions. This is followed by chapters on theta functions, modular groups and modular functions, the quintic, the imaginary quadratic field, and on elliptic curves. The many exercises with hints scattered throughout the text give the reader a glimpse of further developments. Requiring only a first acquaintance with complex function theory, this book is an ideal introduction to the subject for graduate students and researchers in mathematics and physics.
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