Question

Find the residue of the function \[ f(z) = \frac{1}{z^2 + 1} \] at the pole $z = i$.

• A. $\frac{1}{2}$
• B. $\frac{-1}{2}$
• C. 1
• D. $-\frac{1}{2}$

Hint:

To calculate the residue at a simple pole, multiply the function by $(z - z_0)$ and take the limit as $z \to z_0$.

Answer 1

The Correct Option is D

The function has a simple pole at $z = i$. The residue at $z = i$ is given by: \[ \text{Res}(f, i) = \lim_{z \to i} (z - i) f(z) \] Substituting into the function: \[ \text{Res}(f, i) = \lim_{z \to i} \frac{z - i}{z^2 + 1} = \lim_{z \to i} \frac{z - i}{(z - i)(z + i)} = \frac{1}{2i} = -\frac{1}{2} \]