Residue of 1/(z-sin(z)) at z=0

Question

I am to find the residue of f(z)=1/(z-sin(z)) at z=0.
I am confused as to which method to use.
Your help will be greatly appreciated! Thanks!

$$z-\sin z=z-\left(z-\frac16z^3 +O(z^5)\right)=O(z^3)$$Can you find the residue now? — Mark Viola, Oct 28 '15 at 04:54

score 1 · Answer 1 · answered Oct 28 '15 at 12:11

One has $$z-\sin z={z^3\over6}\left(1-{z^2\over20}+?z^4\right)$$ and therefore $${1\over z-\sin z}={6\over z^3}\left(1+{z^2\over20}+?z^4\right)=6{1\over z^3}+{3\over10}{1\over z}+?z\ .$$ It follows that the residue in question is ${3\over10}$.

(Each question mark above represents a certain power series which is convergent in a neighborhood of $0$.)

Residue of 1/(z-sin(z)) at z=0

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