Laurent Expansion of sin(1-1/z)

[nicksauce]

Homework Statement

Find the Laurent expansion of $$f(z) = \sin(1-\frac{1}{z})$$ about z = 0, and state the annulus of convergence.

Homework Equations

The Attempt at a Solution

I tried doing the regular expansion of sin(z), then applying the binomial expansion on the (1-1/z)^n terms, but I can't help but feel that there's a better way to approach the problem. Any thoughts?

 

[HallsofIvy]

I would suggest that you use the sine sum formula to write it as
$$sin(1-\frac{1}{z})= sin(1)cos(\frac{1}{z})- cos(1)sin(\frac{1}{z})$$
and then use the Taylor's series for sine and cosin.

 

[nicksauce]

Boy don't I feel dumb. Thanks!

 

[sarazamani]

is this sentence correct?
sin(1/z)=1/z-1/z^3*3!+1/z^5*5!+-...
for laurent series?