Real Plane, Complex Plane

Date: 09/16/2002 at 19:20:59
From: John
Subject: Complex Analysis

When does Az + Bz + c = 0 become a straight line?

Date: 10/28/2002 at 17:05:48
From: Doctor Nitrogen
Subject: Re: Complex Analysis

Hi, John:

The complex valued equation

   w = f(z) = Az + Bz + c = (A + B)z + c = 0,

is never really a straight line. Note there is a difference between

   w = f(z) = (A + B)z + c, and

   w = f(z) = (A + B)z + c = 0.

And a function w = f(z) in the complex plane does not have the same 
meaning or nature as a real valued function y = f(x) in the real 
plane. So, 

   y = f(x) = (A + B)x + c = 0

in the real plane, does not necessarily mean that

   w = f(z) = (A + B)z + c = 0

will have the same nature or character in the complex plane. The 
reason is that in the real plane you have only two points to 
consider,

   (x, y)

or the value of x in the domain of f, and the value of y in the range 
of f. On the other hand, in the complex plane, you have four values 
to consider:

   (x, y) for x + yi in the complex x-y plane, and

   (u(x, y), v(x, y)) for u(x, y) + v(x, y)i  

in the complex u-v plane, where w = f(z) = u + vi = (A + B)z + c = 0.

Therefore, to elaborate on your question, for the complex valued 
function 

   w = f(z) = (A + B)z + c = 0,

you would have a little "vector" with its terminus at the point x + yi 
in the x-y complex plane, and you would have 

   w = f(z) = u(x, y) + v(x, y)i = (A + B)z + c = 0

in the second, u-v plane.

In the u-v plane, w = f(z) = (A + B)z + c = 0, meaning

 (1)  z = -c/(A + B), and that

      w = f(z) = u(x, y) + v(x, y)i = (A + B)z + c = 0

        = Re((A + B)z + c) + Im((A + b)z + c)i = 0

        = 0 + 0i = 0, because of (1). 

     Hence

     u(x, y) = v(x, y) = 0, in the u-v plane.

But this would just give the origin in the u-v plane, meaning the 
point (x + yi) in the complex x-y plane gets mapped to the origin   
(0, 0) in the complex u-v plane, by w = (A + B)z + c = 0.

Incidentally, in the real plane, the equation 

   y = f(x) = (A + B)x + c = 0

would lead to the point (-c/(A+B), 0)), not the origin. 

Keep in mind there is a difference between

   w = f(z) = (A + B)z + c, and

   w = f(z) = (A + B)z + c = 0, in complex analysis.

Similarly, there is a difference between

   y = f(x) = (A + B)x + c, and

   y = f(x) = (A + B)x + c = 0, in real analysis.

I hope this helped answer the question you had concerning your 
mathematics problem. You are welcome to return to The Math 
Forum/Doctor Math whenever you have any math-related questions.

- Doctor Nitrogen, The Math Forum
  http://mathforum.org/dr.math/