Functions of a Complex Variable

Complex Analysis · Axiom Academy

LESSON Functions of a Complex Variable A function w = f(z) sends the plane to itself — so we picture it as a mapping, not a graph. A complex function is a rule that assigns to each z in a domain D a unique output w . Writing z = x + iy for the input and w = u + iv for the output, the single complex rule splits into two real functions of the two real variables x and y . input z = x + iy → output w = u + iv the real part u and imaginary part v each depend on both x and y Because the input and output are both planes, we cannot draw a single graph. The animation shows the honest picture: a point in the z -plane on the left is carried to its image w = f(z) in the w -plane on the right. Here f(z) = z^2 , and the input z = 1 + 2i lands on w = -3 + 4i . For any specific f , expand f(x + iy) with ordinary complex arithmetic and collect the real and imaginary parts. A few standard functions and their components: The mapping picture explains the shape of a function. Take f(z) = z^2 again and feed it a square grid in the z -plane. The animation carries each grid line to its image: a vertical line x = c and a horizontal line y = c both bend into parabolas in the w -plane (their images satisfy and ). Straight grid lines do not stay straight — the function curves the plane. Maps to a left-opening parabola: . Maps to a right-opening parabola: . Lines parallel to the axes become families of confocal parabolas under .

This is the written version of the interactive lesson above. See the full Complex Analysis course.