Plotting the Image of a Complex Function: w=1/z

[TheFerruccio]

Homework Statement

Find/sketch the image of the function under the transform w = 1/z

Homework Equations

x=-1

The Attempt at a Solution

So, I decided to take the mapping 1/z as 1/(x+iy) For x=-1:

$$\begin{align}<br /> w=\frac{1}{z}&=&\frac{1}{x+iy}&=&\frac{-1-iy}{1+y^2}<br /> \end{align}$$

Getting u in terms of v...
$$u=\frac{-1}{1+y^2}, v=\frac{-y}{1+y^2}$$

To substitute u in for y:
$$\begin{align}\\<br /> (1+y^2)u&=&-1\\<br /> 1+y^2 &=&\frac{-1}{u}\\<br /> y^2&=&\frac{-1}{u}-1\\<br /> y&=&\pm\sqrt{-\frac{1}{u}-1}<br /> \end{align}$$

so...

$$\begin{align}<br /> v&=&\frac{-y}{1+y^2}\\<br /> &=&\frac{\pm\sqrt{-\frac{1}{u}-1}}{\frac{1}{u}}\\<br /> &=&\pm u\sqrt{-\frac{1}{u}-1}<br /> \end{ailgn}$$

This solution of v in terms of u, or the reverse, usually worked for me for finding how to map the curves in the complex plane. However, the book's answer is much simpler, and something that I have no idea how to graph. My main confusion over complex analysis is when to use x, y, u(x,y), v(x,y), and w(z(x,y)).

The answer in the book is:
$$\left|w+\frac{1}{2}\right|=\frac{1}{2}\right$$
And, I have no idea how to graph that.

 

[praharmitra]

Two things first.

1. Were you able to figure how your answer is equivalent to that given in the book?

2. What is the geometrical meaning of |z1-z2| ?