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It's most certainly not obvious how the relation df(x)/dx = f(x), the ratio between the diameter and circumference of a circle , and the square root of -1 -- a "trick" to extend the number line with a second dimension -- are related, let alone be used to create an useful construct.

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Its interesting to see the discrete case.

Let D_n be a difference operator like D_n(F) = F(x+n) - F(x)

Solving by induction for the equation D_n(F)/D_n(x) = F(x), F(0)=1, it gives (1+n)^(x/n).

So now you have a "family" of functions that when "derived" by its difference, gives the same constant ^ x. Euler is the case when n→0.

This aplies for every n, including complex numbers.

Add in the fact that the reason they are related appears to involve infinity (power series). It is tough to argue against it being an impressive result.

I wouldn't say power series are the 'reason' they are related. The more intuitive reason for Euler's identity is the combination of e^x being its own derivative, and the relationship between complex multiplication and rotation.

"its own derivative" is defined in terms of things happening at infinity though. Calculus is very much a study of what happens to things at the infinite and infinitesimal; and e has its heft because of calculus.

OK, agreed.

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