Complex Analysis and Its Applications
Abstract
It is universally acknowledged that “Negative numbers do not have square roots in
the field of real numbers”. Thus we come to two simple deductions: “Only positive
numbers have even power roots in real number field” and “None of even power roots is
negative”. Probably in order to make the conception of “number” more extensive, the
operations more closed, or perhaps to meet the certain demands of the research, dyadic
numbers --- complex numbers --- got their start! The whole field of complex numbers
consists of all the linear combinations of the real number unit “
1” and the imaginary
number unit “
i ” (the arithmetic square root of 1− ). This thesis is mainly a brief
discussion of great benefits brought by the field of complex numbers to the development
of elementary mathematic, including using complex analysis method to prove or solve
some plane geometry problems that are complicated or ingenious, adopting Cauchy
Residue Theorem in complex analysis to figure out some improper Riemann integral in
real number field which is unquantifiable, exploring the difference and linking between
mathematical analysis and complex analysis, and so on.
Key words: Complex Analysis Method, Plane Geometry, Cauchy Residue Theorem,
Improper Integral, Analytical Function, Harmonic Function
