Singular points, poles, residue of z²/((z-2)(z²+1))

Definitions

• Singular point: a point z = a where f(z) is not analytic (it fails to be differentiable there), although every neighbourhood of a contains points where f is analytic. An isolated singular point has a punctured neighbourhood in which f is analytic.
• Pole: an isolated singular point z = a where f(z) = g(z)/(z − a)ᵐ with g analytic and g(a) ≠ 0. The integer m is the order of the pole (m = 1: simple pole). Near a pole, |f(z)| → ∞.

Residues of f(z) = z²/((z − 2)(z² + 1))

Denominator = (z − 2)(z − i)(z + i). The numerator z² does not vanish at 2, i or −i, so there are three simple poles: z = 2, z = i, z = −i.

For a simple pole, Res at a = lim (z → a) (z − a) f(z).

• At z = 2: Res = z²/(z² + 1) = 4/5
• At z = i: Res = z²/((z − 2)(z + i)) = i²/((i − 2)(2i)) = −1/(−2 − 4i) = 1/(2 + 4i) = (2 − 4i)/20 = (1 − 2i)/10
• At z = −i: Res = z²/((z − 2)(z − i)) = (−1)/((−i − 2)(−2i)) = −1/(−2 + 4i) = 1/(2 − 4i) = (1 + 2i)/10

Check: sum = 4/5 + (1 − 2i)/10 + (1 + 2i)/10 = 4/5 + 1/5 = 1, which equals the coefficient of 1/z in f at infinity (f ~ 1/z), as it should.

Answer: Res(2) = 4/5, Res(i) = (1 − 2i)/10, Res(−i) = (1 + 2i)/10.