Integrating Sin(1/z) and Z Sin (1/Z^3) Over C (Circle of Radius 1)

[fahd]

hi there
im confused with this question..
Integrate 1) Sin(1/z) dz
and 2) Z sin (1/Z^3)

where Z is any complex number., over C which is a circle of radius 1 centred at 0

i tried using the cauchy integral formula and stuff but somehow the answer always comes infinity...is that right ..please help..got exam tomorrow!

 

[matt grime]

Where are the poles and what are its residues at each pole?

 

[benorin]

Use Laurent series

 

[HallsofIvy]

Presumably, you know the Taylor's series for sin(z). Replace z in that series by 1/z and 1/z³ to get the Laurent Series for sin(1/z) and sin(1/z³) respectively. Then the integrals should be easy.