a) Evaluate the following contour integrals which are taken anticlockwise around the boundary of the disk |z| = 2 of radius 2 centred at the origin.Ensure you justify your answers. \text { (i) } \int_{|z|=2} \frac{e^{3 z}}{z-i} d z ) If C₁ is the straight line path from 0 to 1, C₂ is the straight line path from 1 to1 + i and C is the straight line path from 0 to 1 + i. i) Show C₁, C₂ and C on a labelled Argand diagram. \text { ii) Using the fact that } \int_{C} f(z) d z=\int_{t=a}^{b} f(z(t)) \frac{d z}{d t} d t \text { find: } \text { (B) } \int_{C_{2}} R e(z) d z Hence find: \int_{C_{1}+C_{2}} R e(z) d z where C₁ + C₂ indicates the total path C₁ followed by C₂. \text { ii) Find } \int_{C} R e(z) d z \text {. What do you conclude about } \int_{C} R e(z) d z \text { ? } \int_{|z|=2} \frac{e^{z}}{6 \pi i-2 z} d z \int_{\mid z=2} \frac{1}{z^{4}} d z \int_{C_{1}} R e(z) d z