A complex variable z has a real part x and an imaginary part y and is written z = x + iy. Functions f(z) of a complex variable are our concern. Explicit examples are 1/z, z2, and exp(z). Any function of a complex variable can be expressed as
where u and v are real functions.
A function f(z) is said to be continuous at z = z0 if given any positive number , one can find a number δ such that |f(z) − f (z0)| < provided that |z − z0| < δ.
Let z0 = x0 + iy0. It then follows that |u (x, y) − u (x0, y0)| ≤ |f(z) − f(z0)| < if |z − z0| < δ i.e., |x − x0| < δ, y − y0| < δ/. Hence, if f(z) is continuous, so is u(x, y) and similarly v(x, y). Conversely, it can be shown that if u(x, y) and v(x, y) are continuous, so ...