The Integration of Functions of a Single Variable, Vol. 2 (Classic Reprint) - Hardcover

G. H. Hardy

9780260190727: The Integration of Functions of a Single Variable, Vol. 2 (Classic Reprint)

Hardcover

ISBN 10: 0260190721 ISBN 13: 9780260190727

Publisher: Forgotten Books, 2018

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Discover how difficult integrals unlock new kinds of functions and how they fit into a larger theory of numbers and curves.

This edition presents the ideas behind algebraical and transcendental functions with a focus on finite form, elliptic integrals, and the role of Liouville’s work in deciding when an integral can be expressed by elementary means. The material is suitable for readers who want a solid, structured approach to advanced calculus and the theory of functions, without requiring deep prior research.

The book surveys how integrals relate to algebraic curves, the notion of deficiency, and the conditions that make certain integrals reducible to known forms. It ties together historical proofs, practical methods, and the limits of current techniques, all explained at a level accessible to motivated readers with a solid undergraduate background.

How certain integrals connect to elliptic and Abelian forms, and what that means for expressibility.
Liouville’s theorems on when integrals are elementary, including the role of exponential and logarithmic terms.
A guided look at when to expect finite-term solutions versus genuinely new transcendents.
Examples that illustrate transforming complex integrals into standard, recognizable forms.

Ideal for readers of advanced calculus, mathematical analysis, and number theory who want a principled, historical, and practical view of how integrals shape the theory of functions. This edition is a solid reference for students and professionals seeking a cohesive account of algebraical and transcendental functions.

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