Complex function

For each point $z = x + yi$ in one region $D$ on the complex plane, one complex number $w = u + vi$, $u = u(x,y), v = v(x,y)$ corresponds, $w$ is called a complex function defined in the region $D$, and $w = f(z) = u + iv = u(x, y) + iv(x, y)$. This region $D$ is called the domain of $f(z)$.

Example 2..1   The domain of $w = f(z) = z^2$ is all plane and $u = x^2 - y^2, v = 2xy$

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 $w = f(z)$ 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 $w$ plane parallel to the real and imaginary axes of the $z$ plane.

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

(a)

$w = z^3$

(b)

$w = \frac{z}{z+1}$

(c)

$w = \frac{z - i}{z + i}$