What's your favorite illustration of the usefulness of complex numbers?
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i mean i can see your point, that nothing is stopping you from making that definition. however, i think it's a little trivial because there is definitely circular reasoning when you find the derivatives of those functions when Euler's identity was built on those derivatives(1/2)
that's simply just a different definition of Euler's identity .. the fact is that it's "neat" and "can be done" is what is the spirit of the OP
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i mean yeah it can be done, however this doesnt make the reasoning valid. the lim x-> 0 of sin(x)/x "can be done" with l'Hôpital, but the reasoning involved with l'Hôpital's rule in this scenario is circular
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