Complex Transformations

Question

Before I ask my question, I'd like that all responses contain little Complex Analysis terminology as possible. This is due to the fact that I'm not actually taking a Complex Analysis course, this is just a Complex numbers + transformations course.

First question: What is the Z-plane and the W-Plane? I thought all complex numbers were mapped in the Argand plane?

Second Question: Why can't we draw complex functions in the Argand plane? It's just a function that outputs complex numbers after all?

Third Question: How do you read this: $$ T:w=\frac{16}{z} $$

Think that is all...

Answer 1

Let's take a step back to real functions: If you have a transformation which maps a real number $x$ to a different real number $y$ via, say, $T: x \mapsto y=\frac{16}{x}$, then we can graph the transformation (or function) in a two-dimensional plane (the $x-y-$plane), since the input $x$ is one-dimensional and the output $y$ is one-dimensional, too.

For complex numbers, if you represent $z$ in the Argand plane, you already have a two-dimensional input (real and imaginary part of $z$) and you map these to a two-dimensional output (real and imaginary part of $w$). Because it's hard to graph in four dimensions, just putting the input and output into the same diagram is hard. One way to overcome this is to ask "how are lines and circles in the $z-$plane mapped into the $w-$plane?"

My interpretation of the terms $z-$plane and $w-$plane is "complex plane of the inputs and outputs, respectively".

Thirdly, I'd read $T:w=\frac{16}{z}$ as a function $T(z)=w=\frac{16}{z}$, i.e. $z \mapsto \frac{16}{z}$.

Answer 2

The graph of a function is a set of ordered pairs of the form (input value, output value), or in this case, $(z,w)$. But $(z,w)$ is a pair of complex numbers, so is not representable as a single element of $\mathbb{C}$. Since each of $z,w$ is an element of $\mathbb{C}$, or in geometric form, an element of $\mathbb{R^2}$, each pair $(z,w)$ is an element of $\mathbb{C}^2$, which can be regarded geometrically as $\mathbb{R}^2{\,\times\,}\mathbb{R}^2$, or alternatively, as $\mathbb{R}^4$, but in any case, would require $4$ real dimensions. Since $4$-dimensional space is not something we humans can visualize, we split the pair $(z,w)$, focusing separately on the behavior of the inputs $z$, and the outputs $w$, each requiring only $2$ dimensions.