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How do you compute the line integral $\int_{C(0,1)} \frac{cos(z)}{z^d} dz$, $d\in\mathbb{Z}$, $z\in\mathbb{C}$? $C(0,1)$ denotes the circumference of radius 1 and center 0 in the complex plane. I don't know how to reduce the integral to an easy one.

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  • $\begingroup$ Have a look at the Cauchy integral formula. $\endgroup$ – Martin R Nov 6 '16 at 14:02
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For $d \leq 0$ the integrand is entire and hence the integral is $0$.

Suppose $d> 0$. The line integral is given by

$$\int_C \frac{\cos z}{z^d} dz = \frac{2\pi i}{(d-1)!} (\cos z)^{(d-1)}|_{z = 0},$$

by the generalised Cauchy integral formula. Here $(k)$ represents the $k$th derivative of the function. From here on just consider cases on $d$.

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