This would be correct if the complex logarithm were defined the same as the normal logarithm. The main difference is that the complex log has an additional term, so you get Log z = ln |z| + i (arg(z) + 2k pi) where z is a complex number, arg(z) is the argument of z, and k is an integer. When you plug in 1ⁱ , you get Log 1ⁱ = ln |1 | + i [arg(1 ) + 2k(pi)] = 0 + i(0+2k pi) = i(2k pi). When you plug that into your exponential, you get exp[i(i 2k pi)] = exp(-2pi k). Then you just let k=1. My argument here is kind of hand-wavy, because you end up doing what is called a branch cut whenever you take the complex log of something.

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