What's your favorite illustration of the usefulness of complex numbers?
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Right, but prove that first equation (over complex numbers), and you'll likely see the derivatives pop out in the process. Depending on the method of proof, of course. I couldn't see the earlier slides from your talk.
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How do you define Exp(ix)?
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the unique function that satisfies the differential equation df/dx = f with initial condition f(0)=1
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