How to calculate Laurent series coefficients?

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I have the following Laurent development for $g(z) = \sin(1/z^3)$ :

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which is defined for all complex except 0.

I have this formula to calculate the coefficient, but we have no example at all how to use it. Just the plain formula (first one is Laurent series, second one is for coefficients) :

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I cannot find a link detailing an example of this calculation.