how do i integrate $ \sin(z^4)/z^4 $ over the unit circle?

Question

how do i integrate $ \sin(z^4)/z^4 $ over the unit circle?

I cannot use cauchy's integral formula since $|z|=1$ and the singularity is at $z=0$ which is not inside the circle.

Is there a way i can integrate this using residues? Thanks.

Answer 1

Is there really a singularity? Look at what happens as $z\rightarrow0$, perhaps by using the Maclaurin series for the sine function.

That being said, technically the function should be written $\operatorname{sinc}(z^4)$.