A Comparison of Different Line, Geometric Representations for Functions of a
Language: English
Published by Forgotten Books, 2024
- Softcover
- New
Seller: Forgotten Books, London, United KingdomForgotten Books
AbeBooks seller since June 2, 2021
Condition: New
US$ 17.10
Quantity: Over 20 available
Add to basketItem description from seller
Print on Demand. This book introduces two methods for constructing a congruence of lines determined by a functional relation between two complex variables, enabling readers to visualize the properties of the function by studying the properties of the resulting congruence. In the first method, two complex variables are represented upon two distinct planes, parallel to each other and a unit apart. If points on the first plane are joined to points on the second plane that correspond to them via a given functional relation, a two-parameter family, or congruence, of straight lines is obtained. The second method uses a Riemann sphere. The two complex variables are projected upon the same sphere, and points of the sphere corresponding to each other via the function are joined by lines. The congruences obtained by either method always have real focal sheets and developables, making them more interesting than those obtained by the first method. The author demonstrates that there are other methods of constructing congruences of lines that impose less drastic restrictions upon the reference planes or spheres. The book then goes on to study the properties of the congruences resulting from these generalizations, including an extension of the method of the Riemann Sphere. By introducing complex variables into the study of lines and geometry, this book sheds new light on a subject of great mathematical interest. This book is a reproduction of an important historical work, digitally reconstructed using state-of-the-art technology to preserve the original format. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in the book.…
Seller Inventory # 9781330190555_0
- Title
- A Comparison of Different Line, Geometric Representations for Functions of a
- Author
- Gladys E. C. Gibbens
- Publisher
- Forgotten Books
- Publication year
- 2024
- Condition
- New
- Book Type
- print-on-demand item
- Binding
- Paperback
- Language
- English
- ISBN 10
- 1330190556
- ISBN 13
- 9781330190555
- Seller catalogs
- Geometry
Explore how line-geometric methods reveal the shape of complex functions.
This dissertation compares different ways to represent functions of a complex variable using line and plane geometry, showing how the same function can look different when you view it through parallel or non-parallel reference planes. It also extends Wilczynski’s ideas by generalizing the Riemann sphere approach to visualize relationships between the two complex variables.
The work frames the study around two main representations. One uses parallel planes to build a congruence from analytic functions, while another allows non-parallel planes, yielding a broader view of the geometric properties. Throughout, the author analyzes how focal points, developable surfaces, and ruled surfaces relate to the underlying functions. The discussion moves toward a general method that connects these geometric pictures to the classic Riemann-sphere idea.
Key ideas include how changing the reference geometry affects the associated coordinates, the conditions under which certain surfaces become developable, and how these viewpoints illuminate common properties of analytic functions. The text also considers simple function examples to illustrate how focal points can be real or imaginary depending on the representation.
- Learn how line and plane configurations encode complex functions and their congruences
- See how parallel versus non-parallel plane setups influence focal structures and developables
- Explore a generalized Riemann-sphere approach to visualize complex-variable relations
- Gain intuition from concrete examples that bridge geometry and complex analysis
Ideal for readers interested in the geometric methods of complex analysis and how different representations shape our understanding of analytic functions.
"Synopsis" may belong to another edition of this title.
Forgotten Books
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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 quality of the aged text.
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