Solution to the problem with Picard

For those who haven’t been following the discussion in the comment thread, here’s the solution to that apparent hole in Picard’s Theorem…

Let us examine the argument slowly:

1. “e^{(e^z)} cannot take the value zero.” This is true. For suppose it did at some point \zeta. Then \alpha=e^\zeta has the property that e^\alpha=0, which we know cannot happen.

2. “Since e^z cannot take the value zero, e^{(e^z)} cannot take the value e^0=1.” This is pure nonsense. It appeals to one-to-oneness that does not hold, spectacularly. Since by Picard’s theorem, e^z only does not take the value zero, it perfectly well can take the value 1756 i\pi (or any other non-zero even multiple of i \pi, which isn’t the year when Mozart was born), and e^{1756 i \pi}=1.


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