Solving Cauchy Residual Theorem for Holomorphic Function at z=2i

[Alekon]

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!

 

[Herick]

Is it 0.00477463?

 

[Herick]

This is $$\frac{\sin (3)}{4 e^2}$$

Let me know if this is the answer. I can explain how i got it.

 

[Alekon]

I finally calculated the answer... it turned out to be -0.2

See the picture if you're interested