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Some Properties of the Exponential Function
We start out by defining the exponential function as the function, ez, where z is a complex number, for which:
These two properties uniquely characterize it.
From property (1.1(a)) it follows immediately that:
Before we go on, we're going to take a short detour for some facts we'll need later.
We haven't done much with complex numbers on this website, but now we need a few details about them. We need to know about complex conjugates, the magnitude of a complex value, and it would be nice to know a little about complex arithmetic. We'll assume the reader is familiar with the basic concept:
And the usual representation of z as a sum of real and imaginary parts:
Addition and multiplication can be worked out mechanically, and are straightforward:
The conjugate of a complex number is formed by replacing i with −i, and is represented with an overline:
Conjugation commutes with arithmetic (as can be easily verified by plug-n-chug):
And if P(z) is a polynomial with real coefficients (like, 3z2 + 2z - 1, for example), then:
The magnitude of a complex number is the distance its location on the complex plain is from the origin, measured in a straight line. It's based on Pythagoras's theorem. It's commonly written with double vertical bars:
It can equally well be defined in terms of the conjugate, which also provides a nice way to compute it:
It commutes with multiplication (but not addition). Compare:
and it's clear that they're they same value.
The most confusing thing about complex numbers may be that they look like pairs of values, but each one is just a single number. The derivative of a complex function at a location is also a single value. It's found the same way we find the derivative of a real function: We take the limit in the change in the function divided by a change in the argument as the change in the argument approaches zero.
The tricky bit is that we need to get the same answer no matter what direction the change is made in. Otherwise the derivative at that point isn't defined. This is a rather tight constraint, in fact, and as a result, the set of differentiable functions of a complex variable is much better behaved (and much smaller) than the set of differentiable functions of a real variable.
And that's about all we need to know about complex numbers for the time
being. We now return to the main event, which is the exponential
We've defined ez as having a derivative of 1 at z=0. We'd also like to find the derivative at an arbitrary point. Assume δz is a small change in z. Then we have:
But we know the derivative of ez is 1 at z=0, so when δz is close to zero we can, to first order, replace eδz with 1 + δz:
Dropping everything higher than first order we find:
where we've used the dotted equals to indicate it's correct only to first order.
Dividing through by δz, and taking limits as we replace our "small changes" with infinitesimal ones, we obtain,
Given that ez is its own derivative, its Taylor series is particularly simple:
It converges everywhere (but we won't prove it), and (even better) it converges to the exponential function everywhere. The exponential function is said to be "analytic" and that's about as far as we're going to go with complex analysis on this site. (Or, anyway, that's as far as we're going with it on this particular page.)
Since ez can be represented as a power series with real coefficients (which is basically just a long polynomial), we can perform some operations on it that don't necessarily work on an arbitrary function. In particular, as with any polynomial with real coefficients, we have:
We'd like to find the magnitude of the value of eiθ. Since, as we observed in section 3, we can find the complex conjugate of the exponential function by conjugating its argument, we can find its magnitude the same way we do for any complex value:
So, eiθ maps the real line onto the unit circle in the complex plane. What can we say about the angle of each point on the circle relative to the real axis?
We know that e0 = 1, and we know its derivative at 0 is 1. Furthermore, we know that the range of eiθ lies in the unit circle, so the path the function follows as θ increases must run along the circle. Therefore, to find the point where eiθ lies on the circle, we need only find the total arc length of the function from e0 to eiθ. So, we want to find:
So, the arc length along the circle from e0 to eiθ is equal to θ. Since arc length along the unit circle is equal to the angle in radians, we conclude from this that the angle (in radians) is equal to θ, which is what we hoped to show.
Well, I think this next approach is more intuitive, anyway. We kind of pulled a rabbit out of a hat when we found the magnitude of eiθ and used that to show that it lies on the unit circle. In this section, we're going to try to show why eiθ remains on the unit circle, just going round and round, as we increase θ.
Consider the equivalence of the complex plane with R2, where the real part becomes the x coordinate, and the imaginary part becomes the y coordinate:
We can treat each complex number as a vector in R2. Consider the effect of multiplication by i. It acts as a rotation operator -- it rotates the vector 90 degrees counter clockwise:
Furthermore, the rotated vector has the same magnitude as the original -- it's just a rotation.
We determined above, in equation 2.6, that ez is its own derivative. So, using the chain rule, we see:
So, the derivative is i times eiθ. Consequently, when we look at the complex numbers as vectors in R2, the derivative of eiθ is perpendicular to eiθ.
So, as θ increases, the vector representing eiθ just rotates -- there is no component of its derivative which lies parallel to it, so it cannot change in length. As a result, it must travel in a circle. So, since e0 = 1, eiθ must lie on the unit circle for all values of θ.
Furthermore, since the derivative has the same magnitude as the vector, and the vector has magnitude 1, the vector must move at a rate of 1 unit of arc length per unit by which θ changes. So, the arc length from e0 to eiθ must be θ.