Change of variables in a complex integral $\int_C^\ \frac{4z} {z^4 +6z^2 +1} dz$

Question

I want to evaluate this integral using Residue Theorem

$$\int_C^\ \frac{4z} {z^4 +6z^2 +1} dz = $$

$$ C : |z| = 1 $$

so I substitute letting $$\ W = z ^ {2 } $$
$$ dw = 2z dz $$

and the contour is $$ s : |w| = 1 $$ and go on normally
$$\int_s \ {} ^\ \frac{2} {w^2 +6w +1} dw $$

and I evaluate that using the theorem normally

the problem is the that the value is half of the correct value....what is the problem then ?

Answer 1

When you make your change of variable you can think of it like going from $\theta \in[0,2\pi]$ to $\theta\in[0,4\pi]$ if you write $z = e^{i\theta}$. Hence you will have a winding number of two and you'll pick up the factor of two that you are missing.