3. Let C be the circle |z| = 2 traversed once in the positive sense. Compute each of the following integrals. (a) ?_C rac{sin 3z}{z - rac{?}{2}} dz (b) ?_C rac{ze^z}{2z - 3} dz (c) ?_C rac{cos z}{z^3 + 9z} dz (d) ?_C rac{5z^2 + 2z + 1}{(z - i)^3} dz (e) ?_C rac{e^{-z}}{(z + 1)^2} dz (f) ?_C rac{sin z}{z^2(z - 4)} dz
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Since the function is analytic everywhere except at $z=0$, and $z=0$ is inside the circle $|z|=2$, we can use the residue theorem. The residue at $z=0$ is given by the limit $$\lim_{z\to 0} z\frac{\sin(3z)}{ze^z} = \lim_{z\to 0} \frac{\sin(3z)}{e^z}.$$ Using Show more…
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