Elements of the Theory of Functions (eBook)
160 Seiten
Dover Publications (Verlag)
978-0-486-16560-8 (ISBN)
This well-known book provides a clear and concise review of general function theory via complex variables. Suitable for undergraduate math majors, the treatment explores only those topics that are simplest but are also most important for the development of the theory. Prerequisites include a knowledge of the foundations of real analysis and of the elements of analytic geometry.The text begins with an introduction to the system of complex numbers and their operations. Then the concept of sets of numbers, the limit concept, and closely related matters are extended to complex quantities. Final chapters examine the elementary functions, including rational and linear functions, exponential and trigonometric functions, and several others as well as their inverses, including the logarithm and the cyclometric functions. Numerous examples clarify the essential ideas, and proofs are expressed in a direct manner without sacrifice of completeness or rigor.
German mathematician Konrad Knopp (1882–1957) taught at the University of Königsberg from 1915-26 and at Tübingen University from 1926 until his retirement in 1950. His other Dover books include Infinite Sequences and Series, Theory and Applications of Infinite Series, Theory of Functions, and Problem Book in the Theory of Functions.
Section I. Complex Numbers and their Geometric RepresentationChapter I. Foundations 1. Introduction 2. The system of real numbers 3. Pointgs and vectors of the planeChapter II. The System of Complex Numbers and the Gaussian Plane of Numbers 4. Historical remarks 5. Introduction of complex numbers. Notation 6. Equality and inequality 7. Addition and subtraction 8. Multiplication and division 9. Derived rules. Powers 10. The system of complex numbers as an extension of the system of real numbers 11. Trigonometric representation of complex numbers 12. Geometric representation of multiplication and division 13. Inequalities and absolute values. ExamplesChapter III. The Riemann Sphere of Numbers 14. The stereographic projection 15. The Riemann sphere of numbers. The point infinity. ExamplesSection II. Linear Functions and Circular TransformationsChapter IV. Mapping by Means of Linear Functions 16. Mapping by means of entire linear functions 17. Mapping by means of the function w = 1/z 18. Mapping by means of arbitrary linear functionsChapter V. Normal Forms and Particular Linear Mappings 19. The group-property of linear transformations 20. Fixed points and normal forms 21. Particular linear mappings. Cross ratios 22. Further examplesSection III. Sets and Sequences. Power SeriesChapter VI. Point Sets and Sets of Numbers 23. Point sets 24. Sets of real numbers 25. The Bolzano-Weierstrass theoremChapter VII. Sequences of Numbers. Infinite Series 26. Sequences of complex numbers 27. Sequences of real numbers 28. Infinite seriesChapter VIII. Power Series 29. The circle of convergence 30. Operations on power seriesSection IV. Analytic Functions and Conformal MappingChapter IX. Functions of a Complex Variable 31. The concept of a function of a complex variable 32. Limits of functions 33. Continuity 34. Differentiability 35. Properties of functions represented by power seriesChapter X. Analytic Functions and Conformal Mapping 36. Analytic functions 37. Conformal mappingSection V. The Elementary FunctionsChapter XI. Power and Root. The Rational Functions 38. Power and root 39. The entire rational functions 40. The fractional rational functionsChapter XII. The Exponential, Trigonometric, and Hyperbolic Functions 41. The exponential function 42. The functions cos z and sin z 43. The functions tan z and cot z 44. The hyperbolic functionsChapter XIII. The Logarithm, the Cyclometric Functions, and the Binomial Series 45. The logarithm 46. The cyclometric functions 47. The binomial series and the general power Bibliography; Index
Erscheint lt. Verlag | 5.10.2016 |
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Reihe/Serie | Dover Books on Mathematics |
Übersetzer | Frederick Bagemihl |
Sprache | englisch |
Maße | 140 x 140 mm |
Themenwelt | Mathematik / Informatik ► Mathematik |
ISBN-10 | 0-486-16560-4 / 0486165604 |
ISBN-13 | 978-0-486-16560-8 / 9780486165608 |
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