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How to evaluate this in the complex plane?

$\int_{\gamma}^{} z^{e^{z^{2}}}\, dz$ when $\gamma$ is the unit circle.

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What about starting with $z=e^{it}$ and then integrate from $0$ to $2\pi$ What have you tried? – draks ... May 10 '12 at 14:46
Are you sure you've typed this correctly? Which branch of the power function are you referring to? Note that $z^w$ is not well-defined without stating this. – mrf May 10 '12 at 15:53
Like mrf said, it is important to choose a branch for the complex logarithm in order for this to be a well-posed problem. However, once this is done I believe that the function in your integral is analytic, so that the integral is zero. – Eric Haengel May 12 '12 at 6:59

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