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What type of singularity does $\exp(\frac{t}{2} (z - \frac{1}{z}))$ have on $z = 0, \infty$ for a fixed $t$? I've concluded this function has simple poles on both $z=0, \infty$ because I thought the function goes to infinity as it approaches to each of those singularity. But I've seen somewhere that they are essential singularities. Which is correct and why?

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    Related: http://math.stackexchange.com/questions/206245/singularities-of-ez-frac1z/206251#2062512012-11-09

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