A Treatise on the Theory of Functions by James Harkness, Frank Morley
A Treatise on the Theory of Functions - Table of Contents
1. Geometric Introduction – Argand Diagram, Fundamental Operations, Strokes, Continuity
2. Real Functions of a Real Variable – Numbers, Sequences, Correspondence of Points and Numbers, Meaning of “Function”, Limits, Continuity
3. The Theory of Infinite Series – Real and Complex Series, Uniform Convergence, Multiple Series, Infinite Products, Integral Series, Circle of Convergence, Cauchy’s Extension, Weierstrass’s Theorem, Behavior at the Circle, Theory of Analytic Functions, Singular Points, Lacunary Spaces, Exponentials and Logarithms, Powers
4. Algebraic Functions – Algebraic and Monogenic Functions, Riemann Surfaces, Examples, Cross-cuts
5. Integration – Integral Functions, Reversion of Series, Lacunary Series, Two-Element Functions, Determination of Coefficients
6. Riemann Surfaces – Canonical Dissection, Klein’s Normal Surface, Delimiting Curves, Branch Points, Integration on the Surface
7. Elliptic Functions – Fundamental Elliptic Function, Liouville’s Theory, Inverse Integral, Applications of Standard Formulae
8. Double Theta-Functions – Addition of Periods, Half-Periods, Rosenhain Hexads, Göpel Relations, Green’s Theorem, Continuation Theorem, Bilinear Relations, Normal Integrals, Abel’s Theorem, Theta-Functions on Riemann Surfaces
9. Dirichlet’s Problem – Boundary-Value Problems for Harmonic Functions
10. Abelian Integrals – Abel Integrals, Applications (Incomplete in Some Editions)
11. Index and Glossary
What You Will Learn in A Treatise on the Theory of Functions
"A Treatise on the Theory of Functions" by "James Harkness" and "Frank Morley" is a comprehensive classical text on "complex analysis" and "analytic functions". It systematically develops the theory of functions of a complex variable, combining geometric intuition with rigorous proofs. The book begins with foundational concepts such as continuity, limits, and the Argand diagram, providing a clear base for understanding more advanced topics.
The treatise explores "power series", Laurent series, meromorphic functions, and essential singularities, as well as analytic continuation and branch-cut analysis. Harkness and Morley also cover conformal mappings, the behavior of functions under transformations, and the geometric aspects of complex functions. Advanced topics such as theta functions, Abel’s theorem, and the theory of periods are also included, offering a deep insight into both classical and modern approaches to function theory.
The book has historically served as a foundational reference for mathematicians and students studying "function theory" and higher-level complex analysis. Its detailed exposition bridges elementary calculus with advanced complex-variable techniques, making it a valuable resource for understanding the structure, behavior, and geometric interpretation of complex functions. This treatise remains influential in the study of analytic functions and the development of modern mathematical analysis.
Book Details & Specifications
Title:
A Treatise on the Theory of Functions by James Harkness, Frank Morley
Publisher:
Macmillan and Co.
Year:
1893
Pages:
536
Type:
PDF
Language:
English
ISBN-10 #:
1164446460
ISBN-13 #:
978-1164446460
License:
Public Domain Work
Amazon:
Amazon
About the Author: James Harkness
The author James Harkness
(1854–1923) and "Frank "Morley (1860–1937) were prominent British mathematicians known for their work in "complex analysis" and "analytic functions". They focused on clear, systematic exposition, combining geometric intuition with rigorous proofs to make advanced concepts accessible to students and researchers. Their collaboration produced "A Treatise on the Theory of Functions", a landmark text covering "function theory", power series, singularities, conformal mappings, and analytic continuation. This work remains influential for its thorough treatment of complex-variable mathematics, bridging elementary calculus with advanced analysis, and shaping how generations of mathematicians study and understand analytic functions and complex analysis.
Read or Downloadable A Treatise on the Theory of Functions
Free Complex Analysis Books PDF | Download University Textbooks