Our goal was to find a reasonable way to expand the exponential function to the domain of complex numbers. We began by demanding the following properties hold when are positive real numbers and are arbitrary complex numbers.
I was curious whether Euler’s famous formula, which students typically see in a college calculus or differential equations course, was useful or meaningful outside the context of calculus.
From those four rules, we deduced that
for some real number .
Then I claimed that
was the simplest choice, which was perhaps unfounded from the standpoint of someone who is not interested in calculus. My justification at the time was that only by choosing to be one can you obtain the simple formula
Outside of calculus, though, there’s no particular reason to believe the hyperbolic and circular trig functions should bear any such relation. At least, no reason that’s completely obvious to me.
An easy way to fix with calculus is to demand
which translates in component form to
The first equality follows from differentiating the trig functions, while the second comes from translating directly into component form.
So fixing comes from the way that is special as an exponent. To see what sort of insight we can get into Euler’s formula without calculus, we’ll need to know in what ways has some special character outside of calculus.
We’ll start with some definitions of . The first one I learned in school was
As my friend Ian recently pointed out to me (while on a long distance run), we can use the binomial theorem, along with the approximation
to find the more practical definition
which is easier to evaluate to a given accuracy. You may also recognize it as a Taylor series expansion of evaluated at .
we can define by
Wikipedia catalogs a remarkable array of alternative definitions to these.
What is still unclear to me is whether this is of any particular interest outside a setting of calculus. may arise when asking a question about compounded interest or certain discrete gambling problems. In those cases arises in the limit as the number of compounds in a unit time or the number of plays of the game increase without bound. That is, arises when we can approximate things as continuous, so that we’ve migrated to the domain of calculus.
The only ways I have on hand to prove the equivalence of all my definitions of are essentially calculus – like. My most direct plan is to substitute
and redefine as
then substitute this into the definition of the natural logarithm to obtain
By allowing to be some arbitrary analytic function of and expanding in a Taylor series
I can make a substitution in the previous formula, reduce it to just one limit, and show that the answer does indeed come to one for arbitrary , but this is far from a calculus-free demonstration, and probably not even right by mathematical standards. I sure wouldn’t want to do a delta-epsilon proof that the method is valid.
There are other ways that are more lucid but less direct and more explicitly rooted in calculus. I could define by its Taylor series, from which it would follow that it is its own derivative, and the other definitions of would fall out naturally enough. Or I might define the logarithm as an integral (a common definition in calculus), show that it’s the inverse function of the exponential of a certain base, and derive the properties of that way. But my point is that none of these make any sense outside of calculus.
That’s why, when I attempted to show Euler’s formula without calculus, I got hung up. Euler’s formula depends on being a special number. Without calculus, it isn’t.
That, at least, is the way things seem to me. The internet is a lot smarter than me, and has a lot more minds. (For example, the internet knows I should technically say, “smarter than I”.) So if does matter when its friends delta and epsilon are on vacation, let me know.
Finally, I’ll clarify a point I made at the end of the previous post, since Nikita rightly asked for more explanation on it.
I examined the behavior of
as . This can be rewritten as
Because it is of the form
it is a point on the unit circle in the complex plane. That is, if you make an “x-axis” with the real part of the number, and a “y-axis” with the imaginary part, your “x-coordinate” is and your “y-coordinate” is . Those points are the points on the unit circle.
But we have
and as gets very small, blasts away to minus infinity. If we imagine
as a dot on the unit circle and slowly start to change the value of , we’ll also change the value of and the dot will move. As becomes close to zero, a small change in produces an enormous change in . In fact, the change in is about , with the change in x.
Allow to count down towards zero with time, starting at with ten seconds on a timer. Then falls smoothly as the time runs out. Instead of watching during this countdown, you watch the point on the unit circle corresponding to . At first it moves just slowly, but in the last second it moves faster, then faster, then faster, until it moves with such rapidity that it flies around the unit circle infinitely many times in the last flitting moment of its existence, and then explodes when and the exponential is undefined.