Question #235403

by contour techniques \int 1/(2+cosθ\theta ) limit 0 to 2π\pi


1
Expert's answer
2021-09-16T00:43:37-0400

02πdθ2+cosθ=02π2dθ4+eiθ+eiθ=02π2eiθdθ4eiθ+e2iθ+1=\int\limits_{0}^{2\pi} \frac{d\theta}{2+\cos \theta}=\int\limits_{0}^{2\pi} \frac{2d\theta}{4+e^{i\theta}+e^{-i\theta}}=\int\limits_{0}^{2\pi} \frac{2e^{i\theta}d\theta}{4e^{i\theta}+e^{2i\theta}+1}=

=2i02πdeiθe2iθ+4eiθ+1=2iz=1dzz2+4z+1=2iz=1dz(z+2+3)(z+23)=\frac{2}{i}\int\limits_{0}^{2\pi} \frac{de^{i\theta}}{e^{2i\theta}+4e^{i\theta}+1}=\frac{2}{i}\oint\limits_{|z|=1}\frac{dz}{z^2+4z+1}=\frac{2}{i}\oint\limits_{|z|=1}\frac{dz}{(z+2+\sqrt{3})(z+2-\sqrt{3})}

Only one pole, z=2+3z=-2+\sqrt{3}, is inside the contour {z=1}\{|z|=1\}, therefore, by the residue theorem we have

02πdθ2+cosθ=2i2πiresz=2+31(z+2+3)(z+23)=\int\limits_{0}^{2\pi} \frac{d\theta}{2+\cos \theta}=\frac{2}{i}2\pi i\,{\rm res}_{z=-2+\sqrt{3}}\frac{1}{(z+2+\sqrt{3})(z+2-\sqrt{3})}=

=4πlimz2+31z+2+3=2π3=4\pi\lim\limits_{z\to -2+\sqrt{3}}\frac{1}{z+2+\sqrt{3}}=\frac{2\pi}{\sqrt{3}}


Answer. 2π3\frac{2\pi}{\sqrt{3}}


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