Elements of the Theory of Functions (eBook)

Konrad Knopp (Autor)

eBook Download: EPUB

2016
160 Seiten
Dover Publications (Verlag)
978-0-486-16560-8 (ISBN)

Infos

Well-known book provides a clear, concise review of complex numbers and their geometric representation; linear functions and circular transformations; sets, sequences, and power series; analytic functions and conformal mapping; and elementary functions. 1952 edition.

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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.

Autor

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.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 5.10.2016
• 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