Exponential Function

Tuji's account

In the beginning, we wanted to multiply number by itself many times. We produced a*a*...*a. Then we made a short hand an where n is a positive integer. After making a-1 to mean the inverse of a and a0 to be the unity 1, we found that an makes sense for all integers n. Then we made sense out of a1/m for positive integer m (at least for positive a) by enforcing that (a1/m)m = a. So far ax is defined for positive real number a and rational number x.

Then all of a sudden people just know that they know what ax is for all positive real number a and real number x. That was a huge leap. Then people went on to differentiate this mysterious object and do various things. I guess, when we learn something at a young age, we just internalise the concept with ease. When one becomes mellower with age and one forgets one's upbringing, one notices that one doesn't really know what real numbers are beside those rational ones. Completion just happened without one really noticing.

And then people wanted ax  for all complex number x and were alarmed that they were asked to swallow Euler's formula e = cos θ + i sin θ and what is e anyway?

Tuji's reconstructed account

We define ex to be the series 1 +  (x/1!) + (x2/2!) + (x3/3!) + ..., which is convergent for all complex number x. One realises that the thing that one has been so familiar with is this unwieldy monstrosity. Then we define ax to be ea log x where log x is the inverse function of ex.

First it is actually quite wieldy. If one differentiates the series term by term (which one can do due to absolute convergence), one gets back the same series! Another property is that ex + y = ex ey which one can see by using the binomial formula.

The trigometric function cos x is defined to be (eix + e-ix)/2 while sin x is defined to be (eix - e-ix)/(2i) for all complex number x. Then the formula e = cos θ + i sin θ (for real number θ) follows directly from definition. Also cos2 x + sin2 x = 1 for all complex number x. This implies that e has modulus 1 for all real number θ. Does this still fit with the geometric picture? At this point the narrator decided to give up. He knows that the this definition of the trignometric functions agree with the geometric definition, but this comes after computing derivatives of sin and cos (from geometric definition) using the addition formulae of sin and cos and make the Taylor expansion. It's not obvious that e2πi = 1 from the serie definition, is it?

Either Euler's formula is trivial and something else is non-trivial or Euler's formula is non-trivial.