1. ## Complex Integration Question

Hi, im having great difficulity with this integration question.

$\displaystyle I=\int_{0}^{2\pi }\frac{d\theta }{ cos\theta + 3 sin\theta - i}$

Thankyou for any help in advance

2. Originally Posted by CeMe
Hi, im having great difficulity with this integration question.

$\displaystyle I=\int_{0}^{2\pi }\frac{d\theta }{ cos\theta + 3 sin\theta - i}$

Thankyou for any help in advance
Recall that you can write
$\displaystyle cos \theta = \frac{1}{2} \cdot ( e^{i \theta} + e^{-i \theta} )$
$\displaystyle sin \theta = \frac{1}{2} \cdot ( e^{i \theta} - e^{-i \theta} )$

Rewrite the integrand using these identities and then substitute $\displaystyle z = e^{i \theta}$ to get an integral over the unit sphere, $\displaystyle \mathbb{S}^1$. From here you can use either Cauchy's theorem or the residue theorem to calculate the integral.

3. ## Complex integration question using residue theorem? -ve or not?

Hi, thankyou for the help, i used residue theorem, however im am unsure if the answer is negative or not. i got
$\displaystyle -2\pi i/\sqrt{11}\\$
Is this correct?
Thankyou for any help in advance

4. Correct answer does not have the negative sign out front. Can you show us more of your work?