Complex Variables

Complex Variables focuses on the principles, characteristics, and functions of complex variables, as well as infinite series, complex numbers, and convergence and divergence. The book first examines complex numbers and the sequences and limits of point sets in the complex plane. Discussions focus on non-decreasing real sequences, boundedness of convergent sequences, boundary points, closed sets, bounded and unbounded sets in the complex plane, complex conjugates, complex numbers as an extension of the real number field, scalar multiplication, modulus, and number pairs. The manuscript then takes a look at the tests for convergence of infinite series, functions of a complex variable, and elementary functions. Concerns cover repeated differentiation of an infinite series, differentiability of power series, hyperbolic functions, link between the exponential and trigonometric functions, orthogonal families of curves, differentiability, testing for convergence or divergence, and series with negative or complex terms. The text examines miscellaneous theorems, contour integration, zeros and singularities, and integration, including order of magnitude of a function, infinite integrals involving trigonometric functions, and sum-limit and anti-differentiation. The publication is highly recommended for students and teachers wanting to explore complex variables.