next up previous
Next: Cauchy-Riemann Relations Up: Two-Dimensional Potential Flow Previous: Introduction

Complex Functions

The complex variable is conventionally written

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

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

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

implies that

$\displaystyle x$ $\displaystyle =r\,\cos\theta,$ (6.4)
$\displaystyle y$ $\displaystyle =r\,\sin\theta.$ (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,

$\displaystyle F(z)$ $\displaystyle = z^{\,2},$ (6.6)
$\displaystyle F(z)$ $\displaystyle = \frac{1}{z}.$ (6.7)

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

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

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

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

then

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

giving

$\displaystyle \phi(x, y)$ $\displaystyle = x^{\,2} - y^{\,2},$ (6.11)
$\displaystyle \psi(x, y)$ $\displaystyle = 2 \,x\,y.$ (6.12)


next up previous
Next: Cauchy-Riemann Relations Up: Two-Dimensional Potential Flow Previous: Introduction
Richard Fitzpatrick 2016-01-22