# Powers!

Quite often, what mathematicians do is generalize from simple ideas to more and more complicated ones, so that what starts out clear and obvious little by little enters the realm of *WTF?* It’s important to remember that the goal isn’t to mess with people; it’s to follow logic wherever it leads without letting “seems crazy” get in the way. If the result is to make people shake their heads and think that understanding math requires being a super-genius, that’s all well and good, but in strict point of fact it’s not true. (Though it would be cruel to tell that to our throngs of beautiful and sexually ravenous groupies; we magnanimously allow them their comforting illusions.)

Exponents are a good example. When they’re positive whole numbers, their meaning is completely straightforward: x^{n} means n x’s multiplied together, so 4^{3} is 4*4*4 or 64. 4^{3} times 4^{2} would be (4*4*4) * (4*4) or 4^{5}; generalizing this, we see that we can multiply by adding exponents. If we now consider the value of x^{0}, we see that x^{0} * x^{n} equals x^{n}, since 0 + n = n. Thus, though this has confused beginning algebra student for centuries, x^{0} must be given the value 1.

Negative numbers aren’t much harder. x^{n} * x^{-n} = x^{0} = 1, so x^{-n} = 1/x^{n} (e.g. 4^{-2}=1/16). Fractions are also straightforward. x^{1/2} * x^{1/2} = x^{1} = x, so

x^{1/2} = √ x e.g. 4^{1/2} = 2 and 4^{5/2} = 32. Irrational exponents are a bit tricker, but still clear. The sequence

x^{3}, x^{3.1}, x^{3.14}, x^{3.141}, x^{3.1415}, x^{3.14159} …

is increasing and bounded above by x^{3.2}, so it has a limit, and we call the value of that limit x^{π}. What else could you do? (Particularly if you want f(x) = a^{x} to be continuous.)

Now that we’ve defined x^{n} for all real n, we’re ready to think about complex n. And there are a few things we’re ready to say:

x^{2i} = x^{i} * x^{i}

x^{-i} = 1/x^{i}

x^{1+i} = x * x^{i}

But none of them help us to define x^{i} in the first place. We need to do some sideways thinking to find the right approach. Let’s see …

Our favorite number to raise to powers is e, because f(x) = e^{x} has the nice property of being its own derivative. (In general, the derivative of a^{x} is a^{x}ln(a), so it’s best to choose the a where ln(a) is 1. Likewise, we let the arguments to sin(x) and cos(x) be in radians so that sin'(x) is simply cos(x), not (π/180)cos(x), as it would be if x were in degrees.) The first derivative of g(x) = e^{-x} is, of course, -e^{-x} or -g(x), by a simple application of the chain rule. Again by the chain rule, the derivatives of h(x) = e^{ix} would be:

First: (i)h(x)

Second: -h(x)

Third: (-i)h(x)

Fourth: h(x)

The second derivative being the negative of the original function should ring a bell. The derivatives of sin(x) are:

First: cos(x)

Second: -sin(x)

Third: -cos(x)

Fourth: sin(x)

So perhaps sin(x) is somehow related to e^{ix}. Let’s look at the power series expansions.

Because e^{x} is its own derivative, and e^{0}=1, its power series is very simple:

e^{x} = 1 + x + x^{2}/2! + x^{3}/3! + x^{4}/4! + …

sin(x) and cos(x) are almost as simple. Recall that their values and derivatives at 0 have the pattern 0, 1, 0, -1, 0, 1, … (cos(x) starting at 1 and sin(x) at 0), so

cos(x) = 1 – x^{2}/2! + x^{4}/4! – x^{6}/6! + …

sin(x) = x – x^{3}/3! + x^{5}/5! – x^{7}/7! + …

Ordinarily we think of x in these expansions as being real. But if we use them to define these functions for complex x too, there are some nice consequences:

- They converge for all complex x, because factorials grow so much faster than exponentials do, so we’re defining them over the whole complex plane.
- It’s easy to verify that e(x) is still its own derivative, but now over the whole complex plane.
- It’s similarly easy to verify that sin'(x) = cos(x) over the whole plane.

Given that, let’s look at what happens if we apply the expansion of e^{x} to ix:

e^{ix} = 1 + ix – x^{2}/2! – ix^{3}/3! + x^{4}/4! + …

or, rearranging a bit (which we can do because the series converges absolutely)

e^{ix} = (1 – x^{2}/2! + x^{4}/4! – …) + i(x – x^{3}/3! + x^{5}/5! – …)

or

e^{ix} = cos(x) + (i)sin(x)

And this is true for all x, by simple algebra. If we restrict x to real numbers, so that we already know the value of the right-hand side, we get a picture of e^{x} when x is a pure imaginary number:

- e
^{0}= 1 (of course) - e
^{x}has a period of 2πi, so that e^{2πi}= e^{4πi}= e^{6πi}= e^{-2πi}= … = 1 - Similarly e
^{πi}= e^{3πi}= e^{5πi}= e^{-πi}= … = -1 - Similarly e
^{πi/2}= e^{5πi/2}= … = i, while e^{3πi/2}= e^{7πi/2}= … = -i

And since the rules for multiplying by adding exponents still apply, e^{a+bi} (where a and b are real) equals e^{a} * (cos(b) + (i)sin(b[/efn_note]. So you can picture (if you’re good at visualizing things in four dimensions) that, along the imaginary axis, e^{x} is a sort of three-dimensional sine curve (it curves in both of the other two dimensions) always lying on a cylinder of radius 1. Along the line x = k + yi that lie parallel to the imaginary axis, (k fixed, y varying), it has the same shape, but gets bigger for positive k and smaller for negative k (the cylinder’s radius being e^{k}.) Along the real axis, e^{x} has the familiar exponential shape. Along a parallel line x = y + ki, (k fixed, y varying), it has the same shape, but rotated according to the value of e^{ki}. (Of course, it rotates back to its familiar location whenever k is a multiple of 2π.)

If that’s too much to try to see in your mind’s eye (and the best I ever do is to frustratingly get almost there), at least you have the most commonly stated consequence of all this:

e^{πi} = cos(π) + i(sin(π[/efn_note] = -1

Euler’s formula!! Not something I’d expect to find here necessarilyReport

nobody expects

~~the Spanish inquisition~~Euler’s formulaReportNowadays it’s called “Titan’s formula”.Report

Thanks, Mike!

While it’s not exactly a Homer doh headslap moment, the end result with the trig functions isn’t terribly surprising to me. Quite the opposite in retrospect, given that we’re playing on the complex plane.

I could only give it a quick read tonight given family interruptions but I’ll work through it in more detail tomorrow.

And again, I really appreciate you humoring my curiosity.Report

It’s important to remember that the goal isn’t to mess with peopleI call bullshit.Report

If that’s too much to try to see in your mind’s eye (and the best I ever do is to frustratingly get almost there)…I’m imagining

f(z, t) =e^(z+it) as a parametric describing a circularly polarized photon traveling through an exponentially expanding universe.I think…Report

Oops… Actually, the wavelength isn’t increasing with time. It’s the amplitude that’s increasing exponentially. Somethin’s blowing up, that’s for sure!Report

Your phasors are stunning, Mr. Schilling.Report

Just a reminder, for those who gave up on Math in high school, all the exclamation marks in this:

is the factorial, not Mike getting really excited about Math. The factorial works like this:

4! = 1*2*3*4 = 24

6! = 1*2*3*4*5*6 = 720Report

I do not know what is worse.

That i have taken the time to read it all, or that i actually liked the article.

Damned engeenering years!Report

These series are incredibly powerful. (Pun semi-intended.) In his classic book, Real and Complex Analysis, using nothing but their power series, Walter Rudin derives every imaginable property of the exponential, sine and cosine functions (both the real and complex versions) in three pages of text.Report

The impressive thing was when he used the power series expansion to prove that cosine, compared to sine, has two extra letters.Report