I still don't quite fully understand about the order of poles and calculating residues.(adsbygoogle = window.adsbygoogle || []).push({});

Take f(z) = 1/sin z at z=0 for example.

When I try putting that into Laurent expansion about z=0,

1/ sin z = 2i/ (e^z - e^(-z)) = 2ie^(-z) / ( 1 - e^(-2z))

= 2ie^(-z) [ 1 - e^(-2z) + e^(-4z) - e^(-6z) + ...] using geometric series

= 2i [ e^(-z) - e^(-3z) + e^(-5z) - e^(-7z) + ... ]

so when I expand out those "e"s using e^(g(z)) = sum_{i>=0} (g(z))^i / i!,

I get no "negative exponent" for z since they're all positive

but then lim_{z -> 0} f(z) = infty so z=0 has to be a pole, right? So then how do we find out the order (of that pole) via Laurent expansion, and consequently how to find its residue at z=0 (which should be equal to 1...)?

Can anyone explain what's going on?

Also, as another example, take f(z) = 1/sin(z^2) at z=0.

Same problem as before...

f(z) = 2i / (e^(z^2) - e^(-z^2))

= 2i [ e^(-z^2) - e^(-3z^2) + e^(-5z^2) - ... ]

How to use the Laurent exansion to find the order of the pole and the residue?

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# Order of a pole

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