Laurent expansion of 1/Sin[z] at z == 2 Pi - Wolfram|Alpha
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Series expansion at z=2 π:

link to /input/?i=1%2F%28-2+%CF%80%2Bz%29%2B1%2F6+%28-2+%CF%80%2Bz%29%2B7%2F360+%28-2+%CF%80%2Bz%29%5E3%2B%2831+%28-2+%CF%80%2Bz%29%5E5%29%2F15120&lk=1&a=ClashPrefs_*Math-
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Approximations about z=2 π up to order 3:

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Series representation at z=2 π:

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