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| Some features of derivative of complex function | |
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77 Some features of derivative of complex function with respect to complex variableS. B. Karavashkin and O.N. Karavashkina Special Laboratory for Fundamental Elaboration SELF e-mail: selftrans@yandex.ru , selflab@mail.ru This paper is the introducing for a monograph devoted to the new branch of theory of complex variable – non-conformal mapping. This new original method enables to connect the mathematical models to which the linear modelling is applicable with nonlinear mathematical models, i.e. with the cases when the mapping function is not analytical in a conventional Caushy – Riemann meaning but is analytical in general sense and has all the necessary criterions of the analyticity, except of the direct satisfying to the Caushy – Riemann equations. As an example, the exact analytical solution of the Bessel-type equation in the continuous range of an independent variable has been obtained. Classnames by MSC 2000: 30C62; 30C99; 30G30; 32A30. Keywords: Theory of complex variable, Non-conformal mapping, Quasi-conformal mapping, Bessel functions
This paper, with all its outward simplicity and obvious statements, is an effort to take a look at the complex plane and operations in it from some unexpected point of view. Or rather, not so much the approach will be unexpected as the concept of complex function will be broadened up to the limits related to most general definitions. First of all, let us state these definitions: “We say that on the set M of the plane Z there has been given the function |
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if the law was given after which to each point z of M a definite point either assemblage of points W was put in correspondence” [1, p. 17]. “When z = x+ iy and w = u+ iv has been given, then to give the function of complex variable w = f ( z ) will mean the same as to give two functions of two real variables, |
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[1, p.17]. As we can see from these definitions, the most general concept of the function of complex variable is not restricted by some before-stipulated direct relation 78 between the real variables u(x, y) and v(x, y) . In particular, |
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etc. also are the functions of complex variable, because to put (3) and (1) into correlation, it is sufficient to represent x and y as |
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