I have the transfer function: $$H(z) = \frac{z^2 + 0.75z + 0.125}{z^2+0.5625}, |z| > 0.75 = \frac{(z-0.5) (z-0.25)}{(z - 0.75j) (z + 0.75 j)}$$
I attempted partial fraction expansion in order to use my lookup table for z-transformations. The issue is my poles are complex conjugate pairs. I have never had to deal with a case like this before and I am not sure how to.
I looked some stuff online and worked out I need to put it into the form of: $$\frac{A_1}{(z - 0.75j)} + \frac{A_2}{(z + 0.75 j)}$$
Doing that,
$$A_1 = (z - 0.75j)\frac{(z-0.5) (z-0.25)}{(z - 0.75j) (z + 0.75 j)} {\Huge\vert}_{z=0.75j} = -0.375 + \frac{0.0833333}{j} + 0.375j $$
$$A_2 = (z - 0.75j)\frac{(z-0.5) (z-0.25)}{(z - 0.75j) (z + 0.75 j)} {\Huge\vert}_{z=0.75j} = -0.375 - \frac{0.0833333}{j} - 0.375j $$
Which does not help me get it into any form on the lookup table.
Looking up more stuff online shows people use polar form which is neat but does not really help me understand what I need to do to get it to use the table.