The Existence of the Exponential Function
The purpose of this paperlet is to use some homological algebra in order to prove the existence of a power series [math]\displaystyle{ e(x) }[/math] (with coefficients in [math]\displaystyle{ {\mathbb Q} }[/math]) which satisfies the non-linear equation
| [Main] |
as well as the initial condition
Alternative proofs of the existence of [math]\displaystyle{ e(x) }[/math] are of course available, including the explicit formula [math]\displaystyle{ e(x)=\sum_{k=0}^\infty\frac{x^k}{k!} }[/math]. Thus the value of this paperlet is not in the result it proves but rather in the story it tells: that there is a technique to solve functional equations such as [Main] using homology. There are plenty of other examples for the use of that technique, in which the equation replacing [Main] isn't as easy. Thus the exponential function seems to be the easiest illustration of a general principle and as such it is worthy of documenting.
