At the start of my first year of university in a Foundations module, my lecturer stated Euler's formula for complex numbers, $e^{i \theta} = \cos\theta + i \sin\theta$. In passing he mentioned this is just a definition which conveniently works when you plug the numbers into the Taylor expansions and that you could actually define it differently (a number other than $e$). What would be the consequences of this? Is there any other number which gives interesting results?
Answer
Suppose you want the following desirable properties:
$\operatorname{exp}:\theta \mapsto e^{i\theta}$ is smooth
$\exp(x+y)=\exp(x)\cdot\exp(y)$
the first requirement is reasonable since we want the function to be nice, and the second is desirable, because it says most essentially that if we add angles,
we get
$(\cos(x)+i\sin(x))(\cos(y)+i\sin(y))$
which after working out with some trigonometric identities reduces to
$\cos(x+y)+\sin(x+iy)$
so that adding angles does what we would expect on the unit circle.
For any function with these two properties: $f(0)=1$ and differentiating with respect to $x$ and letting $x=0$ gives
$$f^{\prime}(y) =f(y)\cdot f^{\prime}(0)$$
which solving returns an exponential function $Ce^{a \theta}$ where $a=f^{\prime}(0)$, although the requirement that $f(0)=1$ forces that $C=1$.
In this case, one could in principle define exponential functions with different bases, but this seems to me inconvenient.
The special thing about base $e$ is that the derivative at $0$ is precisely $1$, which is why we get the clean formulas.
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