Complex Functions

The complex variable is conventionally written

$$z = x + {\rm i}\,y,$$ (6.1)

where ${\rm i}$ represents the square root of minus one. Here, $x$ and $y$ are both real, and are identified with the corresponding Cartesian coordinates. (Incidentally, $z$ should not be confused with a $z$ -coordinate: this is a strictly two-dimensional discussion.) We can also write

$$z = r\,{\rm e}^{\,{\rm i}\,\theta},$$ (6.2)

where $r=\sqrt{x^{\,2}+y^{\,2}}$ and $\theta=\tan^{-1}(y/x)$ are the modulus and argument of $z$ , respectively, but can also be identified with the corresponding plane polar coordinates. Finally, Euler's theorem (Riley 1974),

$${\rm e}^{\,{\rm i}\,\theta}= \cos\theta+{\rm i}\,\sin\theta,$$ (6.3)

implies that

$$x =r\,\cos\theta,$$ (6.4)

$$y =r\,\sin\theta.$$ (6.5)

We can define functions of the complex variable, $F(z)$ , in the same way that we define functions of a real variable. For instance,

$$F(z) = z^{\,2},$$ (6.6)

$$F(z) = \frac{1}{z}.$$ (6.7)

For a given function, $F(z)$ , we can substitute $z=x +{\rm i}\,y$ and write

$$F(z) = \phi(x, y) + {\rm i}\,\psi(x, y),$$ (6.8)

where $\phi(x,y)$ and $\psi(x,y)$ are real two-dimensional functions. Thus, if

$$F(z) = z^{\,2},$$ (6.9)

then

$$F(x + {\rm i}\,y) = (x+{\rm i}\,y)^2 = (x^{\,2}-y^{\,2}) + 2\,{\rm i}\, x\,y,$$ (6.10)

giving

$$\phi(x, y) = x^{\,2} - y^{\,2},$$ (6.11)

$$\psi(x, y) = 2 \,x\,y.$$ (6.12)