Question

Evaluate the integrals below with knowing $f^{(n)}(z)=\frac{n!}{2\pi \,i} \int_C\frac{f(\zeta )}{(\zeta-z)^{n+1}}d\zeta$ . $C$ is the counterclockwise unit circle.

a) $\int_C\frac{\sin z}{z^2}\,dz$

b) $\int_C\frac{\sin z}{z^3}\,dz$

Answer 1

"integral sin z/z^2 dz Here f (z) = sin z, and n = 1 hence integral sin z/z^2 dz = 2 pi i/1! F (z)_z = 0 = 2 pi I times cos z_z = 0 = 2 pi I cos 0 = 2 pi I (b) here f (z) = sin z and n = 2 f' (z) = cos z, f' = (z) = -sin z, hence integral sin z/z^3 dz. = 2 pi/2! F"" (z)_z = 0 2 pi i/2 (- sin z)_z = 0 = pi I sin 0 = 0"