# Thomas Clausen's puzzle [closed]

## closed as off topic by Alain Valette, Kevin Walker, Neil Strickland, Martin Brandenburg, Steven LandsburgMar 17 '12 at 15:22

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The error begins in line zero: Calculate without fixing a definition of $e^z$. –  Martin Brandenburg Mar 17 '12 at 15:09
That is not the problem, $e^z=1+z+{1\over 2}z^2 +{1\over 6}z^3+\ldots$ is perfectly well-defined. –  Robert Kucharczyk Mar 17 '12 at 15:11
The problem is not with defining $e^z$. It's with defining $a^z$ for $a\neq e$. –  Steven Landsburg Mar 17 '12 at 15:26
mathoverflow.net/faq#whatnot –  Yemon Choi Mar 18 '12 at 19:37
There does not seem to be any further need to edit this question. Please let it be. –  S. Carnahan Mar 20 '12 at 8:57

The equality $(e^z)^w = e^{zw}$ does not hold for $z,w \in \mathbb{C}$ in general, so the error lies between lines three and four.
To be more precise, the expression $a^b$ for arbitrary complex numbers $a,b$ is not well-defined, but the expression $e^b$ is. –  Robert Kucharczyk Mar 17 '12 at 15:13