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*e*^{x+iy}: The Imaginary Becomes Real

*That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill-adapted notation. If, for instance, +1, −1, √−1 had been called direct, inverse, and lateral units, instead of positive, negative, and imaginary (or even impossible), such an obscurity would have been out of the question*.—CARL FRIEDRICH GAUSS (1777–1855)^{1}

The introduction of expressions like *e ^{ix}* into mathematics raises the question: What, exactly, do we mean by such an expression? Since the exponent is imaginary, we cannot calculate the values of

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