On Riemann's Theory of Algebraic Functions and Their Integrals by Felix Klein

About this book :-
"On Riemann's Theory of Algebraic Functions and Their Integrals" A Supplement to the Usual Treatises by "Felix Klein" (Translated from the German by Frances Hardcastle) is a classic text that explores the foundations of "algebraic functions", "Riemann surfaces", and "complex analysis". Building on Riemann’s groundbreaking work, Klein presents a clear and structured view of how algebraic functions can be understood through geometric and topological methods. The book focuses on connecting abstract theory with intuitive concepts, making complex ideas more accessible to advanced students and researchers. Klein examines key topics such as "branch points", multi-valued functions, and "elliptic functions", showing how integrals of algebraic functions reveal deep mathematical structures. His approach blends rigorous proofs with geometric intuition, helping readers visualize the behavior of complex functions on Riemann surfaces. The text also explores "Abelian functions", providing a bridge between algebraic theory and complex function theory, which later influenced the development of algebraic geometry and modern analysis. This work remains an important reference for mathematicians interested in the historical and theoretical development of complex function theory. Klein’s careful explanations and systematic presentation make it valuable not only for historical insight but also for understanding advanced concepts in "function theory". The book continues to be cited for its clarity, depth, and lasting impact on the study of algebraic and complex functions.

Book Detail :-
Title: On Riemann's Theory of Algebraic Functions and Their Integrals by Felix Klein
Publisher: Macmillan and Bowes
Year: 1893
Pages: 96
Type: PDF
Language: English
ISBN-10 #: 0486828336
ISBN-13 #: 978-0486828336
License: Public Domain Work
Amazon: Amazon

About Author :-
The author Felix Klein (1849–1925) was a German mathematician known for his work in "geometry", "complex analysis", and "algebraic functions". He developed the Erlangen Program, classifying geometries through symmetry, and greatly influenced modern mathematical thought. Klein focused on making advanced mathematics clear and intuitive, especially in the study of "Riemann surfaces" and elliptic functions. He taught at Leipzig, Göttingen, and Munich, mentoring future mathematicians. His book "On Riemann's Theory of Algebraic Functions and Their Integrals" combines historical insight, geometric intuition, and rigorous analysis, leaving a lasting impact on the study of algebraic and complex function theory.

Book Contents :-
PART I. INTRODUCTORY REMARKS 1. Steady Streamings in the Plane as an Interpretation of the Functions of (x + i y) 2. Consideration of the Infinities of (w = f(z)) 3. Rational Functions and Their Integrals; Infinities of Higher Order Derived from Those of Lower Order 4. Experimental Production of These Streamings 5. Transition to the Surface of a Sphere; Streamings on Arbitrary Curved Surfaces 6. Connection Between the Foregoing Theory and the Functions of a Complex Argument 7. Streamings on the Sphere Resumed — Riemann’s General Problem PART II. RIEMANN’S THEORY 8. Classification of Closed Surfaces According to the Value of the Integer (p) 9. Preliminary Determination of Steady Streamings on Arbitrary Surfaces 10. The Most General Steady Streaming; Proof of the Impossibility of Other Streamings 11. Illustration of the Streamings by Means of Examples 12. On the Composition of the Most General Function of Position from Single Summands 13. On the Multiformity of the Functions; Special Treatment of Multiform Functions 14. The Ordinary Riemann Surfaces over the (x + i y) Plane 15. The Anchor-Ring, (p = 1), and the Two-Sheeted Surface over the Plane with Four Branch Points 16. Functions of (x + i y) Corresponding to the Streamings Already Investigated 17. Scope and Significance of the Previous Investigations 18. Extension of the Theory PART III. CONCLUSIONS 19. On the Moduli of Algebraic Equations 20. Conformal Representation of Closed Surfaces upon Themselves 21. Special Treatment of Symmetrical Surfaces 22. Conformal Representation of Different Closed Surfaces upon Each Other 23. Surfaces with Boundaries and Unifacial Surfaces 24. Conclusion APPENDIX: Glossary of Principal Technical Terms (German ? English)

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