Cauchy integral formula with singularities

Question

I am stuck on this question. $$\int \frac{e^{sin z^{2}}}{(z^{2}+1)(z-2i)^{3}}dz $$ along the path γ where γ is a circle centered at the origin of radius different from 1 or 2. I initially thought of doing a partial fraction decomposition for the denominator but that doesn't seem possible. How can I tackle this question using only the cauchy integral formula and no other advanced theorems? Thank you for reading my query.

Answer 1

You want to use the fact that on such a circle only goes around the poles with radius 1 from the origin corresponding to the term $z^2 + 1$ in the denominator but does not go around the pole with radius 2 from the origin corresponding to the term $(z - 2^i)^3$ in the denominator.

Then using the Cauchy integral formula, you can get a formula for the integral in terms of the function $f(z) = e^{sinz^2}$ in the numerator.

Let me know if you have further questions.

https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula