Calculating a complex line integral over circle radius 4

Question

I am struggling to come to terms with a method of solving some of the problems in vector analysis regarding complex line integrals. $$\int_{|z|=4} \frac{\sin z}{z} dz,$$ $$\int_{|z|=4} \frac{\sin z}{z-5} dz,$$ $$\int_{|z|=4} \frac{\sin z}{z^2} dz.$$ I assume in some way I am supposed to use Cauchy's integral formula but I am not sure, help appreciated.

Answer 1

For the first integral, note that $\frac{\sin(z)}{z}$ has a removable singularity. Then, from Cauchy's Integral Theorem,

$$\oint_{|z|=4}\frac{\sin(z)}{z}\,dz=0$$

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For the second integral, the integrand has a first-order pole at $z=5$. But the region $|z|\le 4$ does not contain $z=5$. Cauchy's Integral Theorem guarantees

$$\oint_{|z|=4}\frac{\sin(z)}{z-5}\,dz=0$$

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For the third integral, the integrand has a first-order pole at $z=0$. Hence, from Cauchy's Integral Formula we have with $f(z)=\frac{\sin(z)}{z}$

$$\lim_{z\to 0}\frac{\sin(z)}{z}=\frac{1}{2\pi i}\oint_{|z|=4}\frac{\frac{\sin(z)}{z}}{z}\,dz$$

Solving for the integral of interest yields

$$\oint_{|z|=4}\frac{\sin(z)}{z^2}\,dz=2\pi i$$