Complex variables logarithmic functions

The definition of many elementary functions can be extended to complex variables. Pol5momial, exponential, and logarithmic functions are discussed here. 

Fromm and Hill s paper, while a sophistieated and almost miraculous application of complex variable theory, produced a formula that exhibited two problems from a practical viewpoint. It contained the dilogarithm function, Li2, and squares of logarithmic functions, in combinations that were multiple-valued with respect to both their real and imaginary parts, and no simple recipe was provided to indicate which branches of these functions should be used. Fromm and Hill s provisional solution was to start from a point in the parameter space where the proper branch was known from asymptotic considerations, and then move in steps to the required parameter values. This procedure was referred to as branch tracking . 

Vincente-Perez, S. Durand, J. S. Alvarez, M. D. 1992. Limitations of complexes logarithmic diagrams as a function of the ligand concentration-diagrams of conditioned variable. An. Quim. 88(7-8) 683-688. 

Let us consider a projection of the complex many-dimensional motion (which variables are both concentrations and the correlation functions) onto the phase plane (iVa, iVb). It should be reminded that in its classical formulation the trajectory of the Lotka-Volterra model is a closed curve - Fig. 2.3. In Fig. 8.1 a change of the phase trajectories is presented for d = 3 when varying the diffusion parameter k. (For better understanding logarithms of concentrations are plotted there.)... 

In this chapter, we discuss symbolic mathematical operations, including algebraic operations on real scalar variables, algebraic operations on real vector variables, and algebraic operations on complex scalar variables. We introduce the concept of a mathematical function and discuss trigonometric functions, logarithms and the exponential function. 

In this chapter we have introduced symbolic mathematics, which involves the manipulation of symbols instead of performing numerical operations. We have presented the algebraic tools needed to manipulate expressions containing real scalar variables, real vector variables, and complex scalar variables. We have also introduced ordinary and hyperbolic trigonometric functions, exponentials, and logarithms. A brief introduction to the techniques of problem solving was included. 

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