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Ocifer
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Homework Statement
[itex] f(z) = \frac{1}{ \exp{ \frac {z^2 - \pi/2}{ \sqrt{3} } } + i } [/itex]
Find the residue of f(z) at [itex] z_0 = \frac{ \sqrt(\pi) }{2 } ( \sqrt(3) - i ) [/itex]
Homework Equations
The Attempt at a Solution
I was able to verify that the given z_0 is a singularity, and furthermore an essential singularity. However, I am stumped at how to figure out the residue (value of [itex] a_{-1} [/itex] ) from the given information.
I've tried rearranging letting f(z) be a general Laurent series, and then rearranged the equation. I've tried making multiplicative arguments, like below:
[itex] \sum_{n \in Z} d_n (z - z_0)^n \cdot ( \exp{ \frac {z^2 - \pi/2}{ \sqrt{3} } } + i ) = 1[/itex]
In the past I've been able to figure out the residue from multiplicative arguments and matching powers, but I don't see how I can do that here. Since the given z_0 is an essential singularity, there will necessarily be an infinite number of negative-degree terms, which must cancel with the infinite number of positive-degree terms from the exponential portion, and it gets so messy that I can't resolve anything.
Furthermore, I am troubled by the fact that I would need an infinite number of negative powers to cancel with an infinite number of positive powers, and so I am not even sure if the usual approach of matching powers will terminate or give an answer.
Can anyone provide a hint, or point out an error I may have made?
I should mention that I've found series for the exponential expression and trivially for the 1, both about z0. But I'm not finding a nice pattern for the coefficients of the exponential so that I can equate coefficients. The first few terms I have but they're not pretty
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