Question

Find residues of f(z)=1/(z^4+1)

Answer 1

All residues are in poles of denominator, other words roots of $z^4 + 1 = 0$.

They are

$$\begin{array}{l} z_1 = e^{i\pi/4} \\ z_2 = e^{3i\pi/4} \\ z_3 = e^{-3i\pi/4} \\ z_4 = e^{-i\pi/4} \\ \end{array}$$

Then:

$$\begin{array}{l} \operatorname{res}\left[f(z_1)\right] = \frac{1}{4z^3} \mid_{z = z_1} = \frac{1}{4} e^{-3i\pi/4} \\ \operatorname{res}\left[f(z_2)\right] = \frac{1}{4z^3} \mid_{z = z_2} = \frac{1}{4} e^{-9i\pi/4} \\ \operatorname{res}\left[f(z_3)\right] = \frac{1}{4z^3} \mid_{z = z_3} = \frac{1}{4} e^{9i\pi/4} \\ \operatorname{res}\left[f(z_4)\right] = \frac{1}{4z^3} \mid_{z = z_4} = \frac{1}{4} e^{3i\pi/4} \\ \end{array}$$