Master Complex Variables Theory and Applications with Kasana PDF: A Book that Introduces the Student

Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician Rafael Bombelli.[18] A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.[19]

Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others. Important work (including a systematization) in complex multivariate calculus has been started at beginning of the 20th century. Important results have been achieved by Wilhelm Wirtinger in 1927.

complex variables theory and applications kasana pdf

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The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example). Unlike real functions, which are commonly represented as two-dimensional graphs, complex functions have four-dimensional graphs and may usefully be illustrated by color-coding a three-dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.

Complex numbers have applications in many scientific areas, including signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis. Some of these applications are described below.

Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the Riemann zeta function ζ(s) is related to the distribution of prime numbers.

In control theory, systems are often transformed from the time domain to the complex frequency domain using the Laplace transform. The system's zeros and poles are then analyzed in the complex plane. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane.

Lang - Complex Analysis (typical Lang style with concise proofs, altough it starts quite slowly, a nice coverage of topological aspects of contour integration, and some advanced topics with applications to analysis and number theory in the end)

I second the answer by "wildildildlife" but specially the book by Freitag - "Complex Analysis" and the recently translated second volume to be published this summer. It is the most complete, well-developed, motivated and thorough advanced level introduction to complex analysis I know. The first volume starts out with complex numbers and holomorphic functions but builds the theory up to elliptic and modular functions, finishing with applications to analytic number theorem proving the prime number theorem. The second volume develops the theory of Riemann surfaces and introduces several complex variables and more modular forms (of huge importance to modern number theory). They are filled with interesting exercises and problems most of which are solved in detail at the end!

"Complex Analysis in Number Theory" by Anatoly Karatsuba.This book contains a detailed analysis of complex analysis and number theory (especially the zeta function). Topics covered include complex integration in number theory, the Zeta function and L-functions. 2ff7e9595c