A lively and vivid look at the material from function theory, including the residue calculus, supported by examples and practice exercises throughout. There is also ample discussion of the historical evolution of the theory, biographical sketches of important contributors, and citations - in the original language with their English translation - from their classical works. Yet the book is far from being a mere history of function theory, and even experts will find a few new or long forgotten gems here. Destined to accompany students making their way into this classical area of mathematics, the book offers quick access to the essential results for exam preparation. Teachers and interested mathematicians in finance, industry and science will profit from reading this again and again, and will refer back to it with pleasure.

Zusammenfassung

R. Remmert and R.B. Burckel

Theory of Complex Functions

*"Its accessibility makes it very useful for a first graduate course on complex function theory, especially where there is an opportunity for developing an interest on the part of motivated students in the history of the subject. Historical remarks abound throughout the text. Short biographies of Abel, Cauchy, Eisenstein, Euler, Riemann and Weierstrass are given. There is an extensive bibliography of classical works on complex function theory with comments on some of them. In addition, a list of modern complex function theory texts and books on the history of the subject and of mathematics is given. Throughout the book there are numerous interesting quotations. In brief, the book affords splendid opportunities for a rich treatment of the subject."-*MATHEMATICAL REVIEWS

Inhalt
Historical Introduction.- Chronological Table.- A. Elements of Function Theory.- 0. Complex Numbers and Continuous Functions.- 1. Complex-Differential Calculus.- 2. Holomorphy and Conformality. Biholomorphic Mappings...- 3. Modes of Convergence in Function Theory.- 4. Power Series.- 5. Elementary Transcendental Functions.- B. The Cauchy Theory.- 6. Complex Integral Calculus.- 7. The Integral Theorem, Integral Formula and Power Series Development.- C. Cauchy-Weierstrass-Riemann Function Theory.- 8. Fundamental Theorems about Holomorphic Functions.- 9. Miscellany.- 10. Isolated Singularities. Meromorphic Functions.- 11. Convergent Series of Meromorphic Functions.- 12. Laurent Series and Fourier Series.- 13. The Residue Calculus.- 14. Definite Integrals and the Residue Calculus.- Short Biographies o/Abel, Cauchy, Eisenstein, Euler, Riemann and Weierstrass.- Photograph of Riemann's gravestone.- Literature.- Classical Literature on Function Theory Textbooks on Function Theory Literature on the History of Function Theory and of Mathematics Symbol Index.- Name Index.- Portraits of famous mathematicians 3.