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Here is a list of some applications of the exponential function.

1) The exponential mapping in Lie theory.

I put this first because my intuition tells me that this must be the most fundamental, or deep, way of thinking about the exponential function. I have often been misled by my intuition however, and the main reason I feel strongly about this is because of how fundamental I consider Lie theory to be.

2) Fourier Series

3) Roots of Unity

4) Gaussian Distribution

5) Boltzmann Distribution

There are certainly other applications, but it always kind of bothered me that I couldn't use symmetry methods to see how (all of) these applications are related. Is it possible that there is no way to do this, i.e. that it's just a happy accident that the exponential function has these applications and it is unrelated to any continuous symmetries?

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(4) and (5) are related to combinatorial limits and the Central Limit Theorem, whereas (1) - (3) are mutually connected in obvious ways. So the challenge comes down to finding a deep link between any of {1,2,3} and {4,5}. One possible avenue of investigation is via the "characteristic function" (Fourier transform) of the Gaussian, which is a key tool in most proofs of the CLT. Another promising route is via the Heat equation, diffusions, and random walks. –  whuber Sep 4 '10 at 18:49
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1 Answer

The most important traits of the exponential function that make $e^x$ occur in that much applications are

a) Euler's identity $e^{i\pi} + 1 = 0$

and its implications for the definition of complex trigonimetric functions, i.e.

$$e^{i \cdot x} = \cos x + i \sin x$$

(like Fourier transformation, unit roots)

b) the fact that $\frac{d}{dx}e^x = e^x$

which lets $e^x$ naturally occur in many solutions of differential equations, which are often the most basic formulation of natural laws (growth/decay, gaussian distribution).

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And using b) $\frac{d}{dx}e^x = e^x$ in taylor series expansion you can get the usual series expression for $e^x$ –  sam Jan 9 '11 at 13:11
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