# Complex Integration Question

• Aug 2nd 2010, 02:07 AM
CeMe
Complex Integration Question
Hi, im having great difficulity with this integration question.

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

Thankyou for any help in advance :)
• Aug 2nd 2010, 02:54 AM
Defunkt
Quote:

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

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

Thankyou for any help in advance :)

Recall that you can write
$cos \theta = \frac{1}{2} \cdot ( e^{i \theta} + e^{-i \theta} )$
$sin \theta = \frac{1}{2} \cdot ( e^{i \theta} - e^{-i \theta} )$

Rewrite the integrand using these identities and then substitute $z = e^{i \theta}$ to get an integral over the unit sphere, $\mathbb{S}^1$. From here you can use either Cauchy's theorem or the residue theorem to calculate the integral.
• Aug 5th 2010, 12:32 PM
CeMe
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
$-2\pi i/\sqrt{11}\\$
Is this correct?
Thankyou for any help in advance
• Aug 5th 2010, 12:35 PM
Ackbeet
Correct answer does not have the negative sign out front. Can you show us more of your work?