In ordinary real analysis, there is this particularly important function called the exponential. Written as exp(x) and defined for all real numbers x, this function so nice because it's the unique solution to the differential equation
d/dx f(x) = f(x)
if we impose the initial value condition f(0) = 1.
With a bit of fiddling, one can discover several other nice properties of the exponential:
- It's equal to its Taylor series expansion at all points.
- It's one-to-one.
- It's positive everywhere.
- It's inverse is actually a logarithm. This particular logarithm is called the natural logarithm, and I will simply write it as log(x), defined for positive x.
Now, these properties imply that exp(x) = e^x where e = exp(1). This number e is the base of the natural logarithm. The caret (^) represents exponentiation. This is where the name comes from.
The point of all this is that something remarkable happens when the exponential is generalized to complex arguments. There's a wonderful theorem in complex analysis that states that, given a real-analytic function defined on the whole real line, there is only one way to generalize this function to the entire complex plane in such a way that
- the generalized function is complex-analytic.
- the generalized function agrees with the original function on the real line.
The term "real-analytic," as mentioned earlier, means the function is equal to its Taylor series at each point of the real line. "Complex-analytic" means that a complex function is equal to its Taylor series (Taylor's theorem extends to complex functions in the natural way) at all points of the complex plane.
The resulting complex exponential function, exp(z) where z is any complex number, is what is of importance for us in studying rotations.
I won't go through the details of this computation--not because it's difficult or complex, but because it's unnecessarily difficult to convey through text without better mathematical typesetting. But it can be shown using a number of clever arguments that we are forced (by the uniqueness of analytic continuation, among other things) to define the complex exponential as
exp(a + i * b) = e^a * [ cos(b) + i * sin(b) ].
If a = 0, we obtain
exp(i * theta) = cos(theta) + i * sin(theta).
Using the geometric meaning of cosine and sine, it's easy to see that exp(i * theta) represents a point on the unit circle in the complex plane at angle theta. It follows that any complex number can be represented in polar coordinates as z = r * exp(i * theta), where r is its distance from the origin and theta is its angle relative to the real axis.
Let's observe what happens when two such numbers are multiplied:
r1 exp(i * theta1) * r2 exp(i * theta2) = r1 r2 exp.</tt><br><br>The magnitudes are multiplied and the angles are added. Given a complex number <tt>z</tt> representing a point in the plane, it follows that we can rotate it about the origin through the angle <tt>theta</tt> by multiplying it by <tt>exp(i * theta)</tt>.<br><br>In case you don't trust that this actually works, it's easy to verify by deriving the classical Cartesian equations for rotation. Suppose <tt>z = x + i * y</tt>. Rotating gives<br><br><tt>z' = z * exp(i * theta) = (x + i * y) * [ cos(theta) + i * sin(theta) ]</tt>.<br><br>Expanding and simplifying,<br><br><tt>z' = x * cos(theta) + x * i * sin(theta) + y * i * cos(theta) - y * sin(theta)</tt><br><tt>z' = [x * cos(theta) - y * sin(theta)] + i * [ x * sin(theta) + y * cos(theta) ].</tt>
