Elementary Functions of Complex Variables

In the course of analysis, elementary functions of the complex variable s arise, such as exp(-s), sin Vs, log(s), and so forth. We must carefully define these operations so that in the limit w - 0, these operations reduce to conventional operations on real variables. We begin our study on the most elementary function, the power law relationship 

We next combine this result with properties of the exp is), which can be proved to have the usual power series representation 

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John continued to overestimate my mathematical skills by persuading me to take an extension course with him. It was enticingly listed as Functions of a Complex Variable and required the purchase of a textbook entitled Conformal Mapping. The syllabus warned that students should not undertake this course without having worked through three years of calculus - John convinced me that in my case one semester of elementary calculus, more than a decade ago, would suffice. 

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

The formal solution to the problem of describing the unsteady-state concentration distribution in a soil column has now been completed. Equation [21] can be used directly for computations using computers that have the software to evaluate the real and complex parts of a complex function such as 7 3/2,t. If such software is not available the real and imaginary parts of 7 3/21 are presented in Appendix 2, in terms of elementary functions that do not involve complex variables, as Eq. [A24] and [A25], respectively. 

The speed of a heterogeneous reaction (or its rate) is usually an extremely complex function that depends on physico-chemical and textural variables (shapes and phase dimensions). (Jenerally, the volumetric or areal speed cannot be defined. Despite thus, a certain nrrmber of heterogeneorrs reactions follow the law , which means that the rate can be written as a product of two functions as in the case of the elementary steps ... 

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