Complex function

For each point $z = x + yi$

$z = x + yi$

in one region $D$

$D$

on the complex plane, one complex number $w = u + vi$

$w = u + vi$

,

$u = u(x,y), v = v(x,y)$

corresponds, $w$

$w$

is called a complex function defined in the region $D$

$D$

, and

$w = f(z) = u + iv = u(x, y) + iv(x, y)$

. This region $D$

$D$

is called the domain of $f(z)$

$f(z)$

.

Example 2..1 The domain of is all plane and

The domain of $w = f(z) = \frac{1}{z}$ is all plane except the origin and $u = \frac{x}{x^2 + y^2}, v = \frac{-y}{x^2 + y^2}$

Since the complex function $w = f(z)$ is considered to be a mapping of the point $z$ on the $z$ plane to the point $w$ on the $w$ plane, the mapping At this time, $w$ is called an image and $z$ is called an original image.

In general, the value of $w$ corresponding to one $z$ is not limited to one, but in my lecture, unless otherwise specified, only one value of $w$ corresponds to a monovalent function.

Exercise2.1

1. For

$w = z^2$

, express

$x,y$

using

$u,v$

function and find out what curve is mapped to the $w$

$w$

plane parallel to the real and imaginary axes of the $z$

$z$

plane.

2. Express $u,v$ as a function of $x,y$ for the following function

(a)
(b): $w = \frac{z}{z+1}$
(c): $w = \frac{z - i}{z + i}$