Complex analysis

Abstract

Complex-valued functions. Two real-valued functions u and v defined in a set M of the x, y-plane assign to every point (x, y) ∈ M a point (u, v) of the u, v-plane. If each of these points (X, y) and (u, v) is regarded as a complex number z = x+ iy and w = u+ ii?, then to every complex number z∈ M there corresponds a complex number w= f(z)= u(x, y)+ iv(x, y). This correspondence represents a complex-valued function f (Fig.). It is called continuous at a point z₀ ∈M if for every sequence {z_n} with z_n ∈M that converges to z₀ for n = 1,2, … the sequence f (z_n) converges to f(z₀). Here a sequence {z_n}, n = 1, 2, …, of complex numbers is said to converge if the sequences {Re z_n} of real parts and {Im z_n} of imaginary parts converge. But this means that f is continuous at z_o = x₀ + iy₀ if and only if u and v are continuous at (X₀, y₀). A function defined in M is called continuous on M if it is continuous at every point of M.