Finding the residues of the following function $\frac{\cos x}{(x^2+a^2)^2}$

Question

I want to find the residues of $f(z)=\dfrac{\cos x}{(x^2+a^2)^2}$

Now it's easy to see that $f(z)$ has 2 poles of order 2 at $z=ia$ and at $z=-ia$

Now by definition, $\displaystyle Res(f,ia)=\lim_{z\rightarrow ia}(z-ia)^2 \dfrac{\cos x}{(z-ia)^2(z+ia)^2}=\lim_{z\rightarrow ia}\dfrac{\cos x}{(z+ia)^2}$ And I don't know how to compute that limit.

Same for $Res(f,-ia)$

Answer 1

Direct substitution gives

$$\lim_{z\to ai}\frac{\cos(z)}{(z+ai)^2}=\frac{\cos(ai)}{(ai+ai)^2}=\frac{\cosh(a)}{-4a^2}$$