Alright so I posted a picture asking the exact question. Here is my best attempt... According to my professor's terrible notes, the numerator can magically turn into the form: e^i(z+3) when converted to complex. The denominator will be factored into (z-2i)(z+2i) but the function is only holomorphic at z=2i so only (z+2i) can be used. From there the Res(f,2i)=g(2i) which is equal to what I believe is something like e^(i(2i+3)/(4i) It follows that J=e^(-2+3i)*Pi and sovling for the real part gives me an incorrect answer. I might be missing some steps but I'm going off a theorem and it's really hard to relate to this problem. Help me PLEASE!