Calculate $\displaystyle \int_\Gamma \frac 1{z^4 + 81}$ where $\Gamma: |z+i| = \frac 34$
Can somebody help me with this question please or give me a hint on how to get started, as I have never seen a question with gamma like this and I have no idea how to start.
Thanks
Solve for $z^4 = -81$, you will get four roots and write it as $\frac{1}{z^4 + 81}=\frac{1}{(z-a)(z-b)(z-c)(z-d)}$ where a,b,c,d are roots of the above $z^4 = -81$. Then use partial fractions $\frac{1}{(z-a)(z-b)(z-c)(z-d)} = \frac{A}{z-a}+\frac{B}{z-b}+\frac{C}{z-c}+\frac{D}{z-d}$. Find the value of A,B,C,D and then substitute in the above equation and then integrate using Cauchys integral formula.