Just a quick one today, as I get caught back up from Thanksgiving. We all know and love the very basic quadratic function. Any second-order polynomial will give you a nice little parabola, which of course is ubiquitous in physics. We all know what that looks like. But what if we're willing to square complex numbers instead of just real numbers? Traditionally we denote complex numbers with z instead of x, so our Sunday Function is:
Ok, so what happens when we square a complex number? Well, we can write any complex number as (a + bi), where "a" and "b" are real numbers. "a" is the real part and "b" is the imaginary part. Keeping in mind that "i" squared is -1, we can go ahead and square our generic expression for any complex number:
The first term (a^2 - b^2) is the real part of the number z^2 and 2ab is the imaginary part of z^2. As such we're done if we just want to calculate numerical values. But we would like a bit better of a theoretical understanding as well. First, we see that the real part is zero if and only if a and b are equal. The imaginary part is zero if and only if one or both of a and b are also zero. So positive real numbers are sent to positive real numbers, imaginary numbers are sent to negative real numbers, and negative real numbers are sent to positive real numbers. Complex numbers will do something in between. In fact if we plot the arg(z^2) [Note: If you think of a complex number as a point on the complex plane, arg(z) represents the angle between the real axis and that point.], we'll get this:
If you think of the complex plane as a rubber sheet, this suggests that the function f(z) = z^2 both stretches the sheet radially and bends it in a counterclockwise direction. To verify this, we'll need to use the polar representation of complex numbers. That'll be a job for next week.
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Nice! 3-D is notable for its chirality. You graphed a right-handed propeller. Fundamental physical theory demands the universe and its mirror image work equally well. The universe disagrees, and increasing so for weaker interactions.
In complex space you see lots of interesting behavior the sheet cut as it is called that shows up in the plot is one example. I wish that gnuplot and more advanced programs had been around when I studied math in the early 70s. More pictures would have helped then.
Kinda cool ... but what's always puzzled me is the infinite number of densely-packed roots you get when you take a root that's irrational. At a guess - the complex number plane gets twisted into an infinitely long spiral by exponentation.
Paul, as I understand it, that is really related at some level to the failure of irrational powers to give you a well-behaved function. f(z)=z^n for positive integers n is trivially holomorphic while f(z)^-n is mereomorphic with a well behaved pole. Even the badness of f(z)^r where r is rational is reasonably well-behaved. But what you get depends closely on what r you pick. So if you want to think of z^a for an irrational a, you can't think of it as a limit of z^r with r being a sequence of rationals approaching z. So the only way to think of z^a is by using the exponential function. And in fact this gives us some insight to what is really going on. Suppose we want to solve for w^a = z. So we really have exp (a log w) = z. Now, assume we have such a w. Then exp(2inPi + a log w) = z for any n. That is, w' will also be a solution if w' = w * exp (log ((2in Pi) /a)) which makes the density issue more apparent because exp (log ((2in Pi) /a) can get arbitrarily close to 1.
This is a rough sketch. I may have screwed up some of the details but the basic idea can I think be made rigorous.
Isn't something like that used to create Julian set or something?
Hi,I'm Iranian. thx alot for gragh of z^2. that help me alot.
This function is really useful for me .I have problem in this kind of example and it is really complicated.I am in weak in mathematics.This one is really useful for me.