3.1 Complex Variable Theory Applied to Two-Dimensional Irrotational Flow

The theory of complex variables is ideally suited to solving problems involving two-dimensional flow. The term complex variable means that a quantity consists of the sum of a real and an imaginary number. An imaginary number is a real number multiplied by the imaginary number . (The terms imaginary and complex distinguish these numbers from real numbers.) A complex number is in fact the sum of two real numbers, the second one being multiplied by the square root of minus one. In many ways complex variable theory is simpler than real variable theory and much more powerful.

Briefly, a complex function F that depends on the coordinates x and y is written in the form

where f and g are real functions. This type of representation has some of the properties of a two-dimensional vector, with the real part standing for the x component and the imaginary part the y component. Thus, the complex number represented by equation (3.1.1) has the directionality properties of the unit vector representation F = f i + g j, (here i and j are Cartesian unit vectors), at least as far as representation and transformation of coordinates is concerned. The two forms of representation differ considerably, however, in operations like multiplication and division.

A complex function F can be represented in graphical form as in Figure 3.1.1, and a spatial...