Calculus of Complex Numbers

Date: 10/10/97 at 16:26:00
From: Yonatan Harel
Subject: Calculus of Complex numbers 

Hello, 

I'm reading this book about chaos, and they describe a method of 
creating a sort of fractal by solving the equation x^4 - 1 = 0 with 
the Newton-Raphson for the plane of complex numbers.

I know the Newton-Raphson for real numbers, but I know nothing of 
differentiating complex numbers, not to mention how to use the 
Newton-Raphson method on an equation with complex numbers.

I will be very greatful if you will explain these things, even if 
they are relatively difficult.

Date: 10/10/97 at 17:48:18
From: Doctor Tom
Subject: Re: Calculus of Complex numbers 

Hello Yonatan,

The subject is called "complex variables" or "complex analysis." It 
has some tricky points, but an amazing amount of what you know from 
the calculus of real numbers continues to work (although this needs to 
be proven, and that's what the subject is about).

For polynomials, the results are almost all the same, and
Newton-Raphson works exactly the same - same derivatives, same 
formula:

     n = o - f(o)/f'(o)

where n is the new guess and o is the old guess. The derivative of 
x^4-1 is 4x^3. What's interesting is that since you're in the complex 
plane, the initial guess doesn't have to be real.

To get the nice fractal, look at the part of the complex plane, say,
where the real and imaginary parts go from -2 to 2. Divide this into a 
couple of hundred steps in both directions, and use each as a starting 
point for Newton-Raphson. If it converges, it will converge to one of 
the four roots:  1, -1, i, or -i. Color the point four different 
colors depending on which root it eventually converges to. You'll get 
a great looking fractal.

Almost any complex polynomial of degree greater than 3 will, with
the same technique, generate an interesting fractal design.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 10/12/97 at 15:39:50
From: Yonatan Harel
Subject: Re: Calculus of Complex numbers

Thank you very much for sending me an answer about the Newton-Raphson 
method with Complex variables.

I would like it very much if you could give me at least a vague idea 
about the actual meaning of derivatives of complex functions.

Thanks again, 

Yonatan

Date: 10/13/97 at 14:40:00
From: Doctor Tom
Subject: Re: Calculus of Complex numbers

They mean exactly the same thing as in real variables.  The best way 
to think of it (for real variables) is as follows.

If f(x) has derivative f'(x), at the point x0, then if you want to 
approximate the curve y = f(x) as accuarately as possible near the 
point x0 with a straight line, the equation that does so is 
y = f(x0) + f'(x0)(x-x0).

Exactly the same thing is true in complex variables, although the 
concept of "line" is different, since the "line" is defined over the 
entire complex plane. Drawing the surface of w = f(z) is equally 
difficult to imagine, since it's also defined over the entire plane, 
and sort of requires four real dimensions to represent. You could draw 
surfaces of the real and imaginary parts, or of the amplitude and 
angle, but you can see it's tough to visualize. There's really no easy 
way (that I know of) to do it.  Just study complex variables for a 
couple of years and you'll start to get an intuition is all I can say.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/