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Some pecularities of derivative of complex function |
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77 Some peculiarities of derivative of complex function with respect to complex variable S. B. Karavashkin Special Laboratory for Fundamental Elaboration SELF 187 apt., 38 bldg., Prospect Gagarina, Kharkov, 61140, Ukraine phone +38 (0572) 276624; e-mail: sbkarav@altavista.com First published in SELF Transactions, vol.1 (1994), pp.77-95 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
With all outward simplicity and evidence of some statements, this paper tries looking from some unexpected standpoint at the complex plane and transformations realising on it. Rather, not so much the viewpoint will be unexpected, but the concept of a complex function will be extended to the frames of the most general definitions. First of all, state these definitions: "It is said that in the set M of points belonging to the plane z the function |
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(1) |
was assigned, if the regularity has been indicated, by which the relation has been set up so that a definite point or points assemblage w was placed in correspondence with each point z of M " [1, p.17]. "If one puts z = x+ iy and w = u+ iv, then to define the function of a complex variable w = f ( z ) will be the same as to define two functions of two real variables |
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(2) |
[1, p.17]. As one can see from the definitions, the most general concept of the function of a complex variable is not limited by some before-conditioned direct relation 78 between the real variables u(x, y) and v(x, y). In particular, the functions |
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(3) |
and so on are also the functions of a complex variable, because to set up the correspondence between (3) and (1), it is sufficient to present x and y in the form |
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Contents: / 77 - 78 / 78 - 79 / 80 - 81 / 81 - 83 / 84 - 86 / 86 - 88 /
