- #1
geft
- 144
- 0
\int_{0}^{2\pi} \frac{d\theta}{5 - 4sin\theta}
= \oint_{C} \frac{dz/iz}{5 - \frac{2z}{i} - \frac{2}{iz}}
= - \oint_{C} \frac{dz}{2z^{2} - 5iz - 2}
= - \oint_{C} \frac{dz}{(4z - 8i)(49 - 2i)}
z_{1} = 2i, z_{2} = \frac{1}{2}
Res = \left|\frac{1}{4z - 2i} \right|_{z = 2i} = \frac{1}{6i}
\Rightarrow 2\pi i(\frac{1}{6i})(-1) = \frac{-\pi}{3}
The answer is 2pi/3, but I can't seem to get it.
= \oint_{C} \frac{dz/iz}{5 - \frac{2z}{i} - \frac{2}{iz}}
= - \oint_{C} \frac{dz}{2z^{2} - 5iz - 2}
= - \oint_{C} \frac{dz}{(4z - 8i)(49 - 2i)}
z_{1} = 2i, z_{2} = \frac{1}{2}
Res = \left|\frac{1}{4z - 2i} \right|_{z = 2i} = \frac{1}{6i}
\Rightarrow 2\pi i(\frac{1}{6i})(-1) = \frac{-\pi}{3}
The answer is 2pi/3, but I can't seem to get it.
)
