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I have been trying to find the residue of $$\frac{-i}{y-\beta i}e^{A + By + Cy^{-1}}$$ in $y=0$. But I'm stuck. I would be really grateful for some help.

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The residue is the coefficient of $z^{-1}$ in the Laurent expansion.

It's not hard to expand $${-i\over y-\beta i}={1/\beta\over1-(i/\beta)y}$$ in a power series in $y$. $e^A$ is just a constant, $$e^{By}=1+By+{(By)^2\over2}+\cdots$$ and $$e^{Cy^{-1}}=1+{C\over y}+{C^2\over2y^2}+\cdots$$ Multiply together, and extract the coefficient of $y^{-1}$. Looks to me like some double series; with any luck, it simplifies at least a bit.

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