- #1
bugatti79
- 794
- 1
Homework Statement
Convert to suitable contour integral and use Cauchy's Residue Theorem to evaluate it
##\int_0^{2\pi} \frac{\sin \theta d\theta}{5-4\sin \theta}##
Homework Equations
Res ##f(z)_{z=z_0} = Res_{z=z_0} \frac{p(z)}{q(z)}=\frac{p(z_0)}{q'(z_0)}##
The Attempt at a Solution
## \displaystyle \int_0^{2\pi} \frac{\sin \theta d\theta}{5-4\sin \theta}=\oint_C \frac{(1/2i)(z-1/z)dz/iz}{5-4(z-1/z)}=\oint_C \frac{(z-1/z)dz}{8z^2-10z-8}##
The denominator has simple poles at ##z_1=5/8+\sqrt{89}/8## and ##z_2=5/8-\sqrt{89}/8## with z_2 being inside the unit circle hence
Res ##f(z)_{z=z_0} = Res_{z=z_0} \frac{z-1/z}{8z^2-10z-8}=\frac{z-1/z}{16z-10}|_{z=5/8-\sqrt{89}/8}##...?