Yuliana Jordan

2022-03-19

Can anyone suggest a simple method to solve this integral by complex variables?

$\int}_{0}^{2\pi}\frac{d\theta}{\omega -a\mathrm{sin}\theta$

where$\left|a\right|<\left|w\right|$ .

where

PCCNQN4XKhjx

Beginner2022-03-20Added 8 answers

The integral is equal to, upon subbing $z={e}^{i\theta}$ and using $\mathrm{sin}\left\{\theta \right\}=\frac{z-{z}^{-1}}{2i}$

$-\frac{2}{a}{\oint}_{\left|z\right|=1}\frac{dz}{{z}^{2}-i2\frac{\omega}{a}z-1}$

The only pole inside the unit circle is at$z=i\left(\frac{\omega}{a}\right)-i\sqrt{{\left(\frac{\omega}{a}\right)}^{2}-1}$ . The integral is then $i2\pi$ times the residue of the integrand at this pole, or

$i2\pi \frac{-\frac{2}{a}}{-i2\sqrt{{\left(\frac{\omega}{a}\right)}^{2}-1}}=\frac{2\pi}{\sqrt{{\omega}^{2}-{a}^{2}}}$

The only pole inside the unit circle is at

Ciara Hoffman

Beginner2022-03-21Added 5 answers

and if we use the substitution

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