Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not' grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory arid the struc ture of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "completely integrable systems", "chaos, synergetics and large-5cale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics. This program, Mathematics and Its Applications, is devoted to such (new) interrelations as exampla gratia: - a central concept which plays an important role in several different mathe matical and/or scientific specialized areas; - new applications of the results and ideas from one area of scientific en deavor into another; - influences which the results, problems and concepts of one field of enquiry have and have had on the development of another.
Inhaltsverzeichnis
1. Introduction. - 2. Definition and Evaluation of Residues. - 3. Contour Integration. - 4. Applications of the Calculus of Residues in the Theory of Functions. - 5. Evaluation of Real Definite Integrals by Means of Residues. - 6. Evaluation of Finite and Infinite Sums by Residues. - 7. Differential and Integral Equations. - 8. Applications of Calculus of Residues to Special Functions. - 9. Calculus of Finite Differences. - 10. Augustin Louis Cauchy. - Notes Added in Proof. - Name Index.
