Evaluating a Cauchy integral $\frac{1}{2\pi}$ $\int_{0}^{2\pi} \frac{d\theta}{(1-2r\cos\theta+r^2)}$

Question

Evaluate

$$\frac{1}{2\pi}\int_{0}^{2\pi} \frac{d\theta}{(1-2r\cos\theta+r^2)}$$ for $0 < r < 1$ by writing ${\cos\theta}$ = $\frac{1}{2}$($ e^{i\theta}+e^{-i\theta})$ and reducing the given integral to a complex integral over the unit circle.

I believe I am supposed to use the Cauchy Integral formula and find the Taylor series centered at $z_0$ but not sure how to proceed

Answer 1

Factorise the denominator. Write the integrand as a product of infinite series, then as a sum of terms of the form $\int r^k e^{li\theta}$ for integers $k,\,l$ with $k\ge 0$. Use $\int_0^{2\pi}e^{li\theta}d\theta =2\pi\delta_{l0}$ (that's a Kronecker delta there). You'll only have a geometric series of common ratio $r^2$ left to worry about.

Answer 2

Since $\int_{0}^{2\pi}e^{ni\theta}e^{-mi\theta}\,d\theta=2\pi \delta(m,n)$ and $$ \frac{1}{1-2r\cos\theta+r^2}=\frac{1}{(1-r e^{i\theta})(1-r e^{-i\theta})}=\left(\sum_{n\geq 0}r^n e^{in\theta}\right)\cdot\left(\sum_{m\geq 0}r^m e^{-mi\theta}\right) $$ we have that $$ \frac{1}{2\pi}\int_{0}^{2\pi}\frac{d\theta}{1-2r\cos\theta+r^2} = \sum_{n\geq 0}r^{2n}=\frac{1}{1-r^2}.$$

Answer 3

Using the substitution $z:=re^{i\theta}$ you can rewrite the integral as $$\frac{1}{2\pi}\int^{2\pi}_0\frac{\,d\theta}{(re^{i\theta}-1)(re^{-i\theta}-1)}=\frac{1}{2\pi i}\oint_{|z|=r}\frac{\,dz}{z(z-1)(\overline{z}-1)}=\frac{1}{2\pi i}\oint_{|z|=r}\frac{\,dz}{(z-1)(|z|^2-z)}$$ which is the same as using $|z|=r$ $$\frac{1}{2\pi i}\oint_{|z|=r}\frac{\,dz}{(z-1)(r^2-z)}$$ Since $0< r<1$ then $r^2<r$ so there is only one simple pole at $z=r^2$ with residue $1/(1-r^2)$. Hence by Cauchy's residue theorem you get $$\frac{1}{2\pi i}\oint_{|z|=r}\frac{\,dz}{(z-1)(r^2-z)}=\frac{1}{2\pi i}\cdot 2\pi i\cdot\text{Res}(\frac{1}{(z-1)(r^2-z)},0)=\frac{1}{1-r^2}$$

Answer 4

Looks like an application of the mean value theorem for harmonic functions $u$ on a region containing the closed unit disk $|z|\leq 1$. Then, for any $|z_0| < 1$ we have:

$$u(z_0) = \frac{1}{2\pi}\int_{0}^{2\pi} u(e^{i\theta})\frac{1-|z_0|^2}{|z_0-e^{i\theta}|^2}\, dz$$

$\frac{1}{2\pi}\int_{0}^{2\pi} \frac{d\theta}{1-2r\cos\theta+r^2} = \frac{1}{2\pi}\int_{0}^{2\pi} \frac{1}{(r-\cos\theta)^2 + \sin^2\theta}\, d\theta \stackrel{z_0 = r,\, u(z) = 1}{=}\frac{1}{2\pi}\int_{0}^{2\pi} 1 \cdot \frac{1-r^2}{|r-e^{i\theta}|^2}\cdot \frac{1}{1-r^2}\, d\theta = \frac{1}{1-r^2}$

Answer 5

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With $\ds{r \in \mathbb{R}\setminus\braces{-1,1}}$:

\begin{align} &\bbox[10px,#ffd]{\ds{{1 \over 2\pi}\int_{0}^{2\pi}{\dd\theta \over 1 - 2r\cos\pars{\theta} + r^{2}}}} = {1 \over 2\pi}\int_{-\pi}^{\pi}{\dd\theta \over 1 + 2r\cos\pars{\theta} + r^{2}} = {1 \over \pi}\int_{0}^{\pi}{\dd\theta \over 1 + 2r\cos\pars{\theta} + r^{2}} \\[5mm] = &\ {1 \over \pi}\int_{-\pi/2}^{\pi/2}{\dd\theta \over 1 - 2r\sin\pars{\theta} + r^{2}} = {1 \over \pi}\int_{0}^{\pi/2}\bracks{% {1 \over 1 - 2r\sin\pars{\theta} + r^{2}} + {1 \over 1 + 2r\sin\pars{\theta} + r^{2}}}\dd\theta \\[5mm] = &\ {2\pars{1 + r^{2}} \over \pi}\int_{0}^{\pi/2}{\dd\theta \over \pars{1 + r^{2}}^{2} - 4r^{2}\sin^{2}\pars{\theta}} \\[5mm] = &\ {2\pars{1 + r^{2}} \over \pi}\int_{0}^{\pi/2}{\sec^{2}\pars{\theta}\,\dd\theta \over \pars{1 + r^{2}}^{2}\sec^{2}\pars{\theta} - 4r^{2}\tan^{2}\pars{\theta}} \\[5mm] = &\ {2\pars{1 + r^{2}} \over \pi} \int_{0}^{\pi/2}{\sec^{2}\pars{\theta}\,\dd\theta \over \pars{1 - r^{2}}^{2}\tan^{2}\pars{\theta} + \pars{1 + r^{2}}^{2}} \\[5mm] = &\ {2 \over \pi\verts{1 - r^{2}}}\int_{0}^{\infty} {\dd t \over t^{2} + 1} = \bbx{{1 \over \verts{1 - r^{2}}}\,,\quad r \in \mathbb{R}\setminus\braces{-1,1}}\quad \mbox{with}\quad t \equiv {\verts{1 - r^{2}} \over 1 + r^{2}}\,\tan\pars{\theta} \end{align}