Abstract
In this chapter, we derive results which are fundamental in the study of analytic functions. These results constitute one of the pillars of mathematics and have far-ranging applications. Notice that many important properties of analytic functions are very difficult to prove without use of complex integrations. For instance, the existence of higher derivatives of analytic functions is a striking property of this type. There occur real integrals in applications that can be evaluated by complex integration. We now turn our attention to the question of integration of complex valued function.
If there is a God, he’s a great mathematician
Paul Dirac
God does not care about our mathematical difficulties;
He integrates empirically
Albert Einstein
Logic and mathematics are nothing but specialized
linguistic structures
Jean Piaget
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Notes
- 1.
Historically, Euler was the first to obtain the value of a definite integral by replacing the variable from real to complex. P.S. Laplace (1749–1827) investigated (1782, 1810) the validity of this Process. S.D. Poisson (1781–1840) is believed to have been the first to use a line integral in the complex plane.
- 2.
Ahlfors uses “regular arc” in the more restricted sense that \(\dot{x}(t)\) and \(\dot{y}(t)\) do not vanish simultaneously.
- 3.
Edouard Goursat(1858–1936), French mathematician. Cauchy published the theorem in 1825. The removal of that condition Goursat (see Transaction of the American Mathematical Society, Vol. (1) (1900), 14–16.) is quite important, for instance, in connection with the fact that derivatives of analytic functions are also analytic. Goursat also made basic contribution to partial differential equations.
- 4.
Giacinto Morera (1856–1909), Italian mathematician who worked Genoa and Turin.
- 5.
Simeon Denis Poisson (1781–1840), French mathematician and physicist, Professor in Paris from 1809. His work includes potential theory, partial differential equations, and probability.
- 6.
This result was presented by Liouville in his lectures in 1847. E.T. Copson is of opinion that result seems to be originally due to his famous contemporary A.L. Cauchy whose Exercises d’analyse, giving his systematic account of the Theory of Functions, appeared in Paris in 1841.
- 7.
Brook Taylor (1685–1731), English mathematician, who introduced this formula for functions of a real variable.
- 8.
The present theorem extends Taylor’s classical theorem in real analysis to a class of analytic functions of a complex variable was, in the words of Prof. W. F. Osgood, “Cauchy’s crowning discovery of 1831”. It may be noticed that this result was obtained by him when he was in political exile in Italy.
- 9.
Colin Maclaurin (1698–1746), Scots mathematician, professor at Edinburgh.
- 10.
Pierre Alphonse Laurent (1813–1854), French engineer and mathematician, published the theorem in 1843.
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Pathak, H.K. (2019). Complex Integrations. In: Complex Analysis and Applications. Springer, Singapore. https://doi.org/10.1007/978-981-13-9734-9_3
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DOI: https://doi.org/10.1007/978-981-13-9734-9_3
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