Complex Analysis: Theory and Applications · Problem 1.2.2

Complex Numbers — Howell & Matthews

Mathematics — Complex Analysis · complex-arithmetic-evaluation

Problem

Evaluate the following quantities. (a) (1+i)(2+i)(3+i)‾\overline{(1+i)(2+i)(3+i)}. (b) (3+i)/(2+i‾)(3+i)/(\overline{2+i}). (c) Re[(i−1)3]\text{Re}[(i-1)^3]. (d) Im[(1+i)−2]\text{Im}[(1+i)^{-2}]. (e) 1+2i3−4i−4−3i2−i\dfrac{1+2i}{3-4i} - \dfrac{4-3i}{2-i}. (f) (1+i)−2(1+i)^{-2}. (g) Re[(x−iy)2]\text{Re}[(x-iy)^2]. (h) Im ⁣(1x−iy)\text{Im}\!\left(\dfrac{1}{x-iy}\right). (i) Re[(x+iy)(x−iy)]\text{Re}[(x+iy)(x-iy)]. (j) Im[(x+iy)3]\text{Im}[(x+iy)^3].

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