Using residues to evaluate the integral $\int_{-\pi}^{\pi} \frac{\cos(n\theta)}{1-2a\cos(\theta)+a^2}d\theta$, $|a|<1$

Question

Calculate the integral for $\left|a\right|<1$

$$\int_{-\pi}^{\pi} \dfrac{\cos(n\theta)}{1-2a\cos(\theta)+a^2}d\theta$$

I'm supposed to evaluate this using method of residues, but the parameter a is confusing me. I'm not even sure what contour I should use. Any help would be very appreciated.

Answer 1

use the unit circle as the contour, so that we have $z=e^{i\theta}$ with $d\theta = \frac{dz}{iz}$

since $\cos n\theta = \Re z^n$ we require $\Re I$ where:

$$ I = \int_{|z|=1} \frac{z^n}{1-a(z+\frac1{z})+a^2}\frac{dz}{iz} \\ $$ $$ = -\frac1{ia}\int_{|z|=1} \frac{z^n}{z^2-(a+\frac1{a})z+1}dz $$

$$ = -\frac1{ia}\int_{|z|=1} \frac{z^n}{(z-a)(z-\frac1{a})}dz $$ since $|a| \lt 1$ only the residue at this point is required which is easily seen to be $\frac{a^n}{a-\frac1{a}}$

hence $$ I = -\frac1{ia} 2\pi i \frac{a^n}{a-\frac1{a}} = 2\pi \frac{a^n}{1-a^2} $$ since $I$ is already real, this is the required value of the original integral

Answer 2

Look at the integral $$ I_n = \int_{-\pi}^{\pi} \frac{e^{in\theta}}{(1-ae^{i\theta})(1-ae^{-i\theta})} \, d\theta. $$ It is easy to see the denominator expands to $1-2a\cos{\theta}+a^2$.

Since the interval of integration is symmetric, we can add the integral with $\theta \mapsto -\theta$ to $I$, and we find the integrand is $$ \frac{2\cos{n\theta}}{1-2a\cos{\theta}+a^2}, $$ so $I_n$ is equal to your integral. Now set $e^{i\theta}=z$, so the interval of integration transforms to the circle $|z|=1$, and $dz/z = i d\theta$. Hence $$ I_n = \frac{1}{i}\int_{|z|=1} \frac{z^{n}}{(1-az)(z-a)} \, dz. $$ Now you do the residues bit: since $|a|<1$, the only pole inside the unit circle is at $z=a$, where the residue is just $$ \left. \frac{z^n}{1-az} \right\rvert_{z=a} = \frac{a^n}{1-a^2} $$ Now we do $I_n = 2\pi i \sum \text{Res}$, and so $$ I_n = \frac{2\pi i}{i} \frac{a^n}{1-a^2} = \frac{2\pi a^n}{1-a^2}. $$

Answer 3

Let's let $z=e^{i\theta}$ so that $dz=i\,e^{i\theta}d\theta$. This complex plane contour $C$ is the unit circle. Thus,

$$\begin{align} \int_{-\pi}^{\pi}\frac{\cos (n\theta)}{1-2a\cos \theta+a^2}d\theta&=\frac12\,\oint_C \frac{z^n+z^{-n}}{1-a(z+z^{-1})+a^2}\frac{dz}{iz}\\\\ &=\frac{i}{2a}\,\oint_C \frac{z^n+z^{-n}}{(z-a)(z-1/a)}dz\\\\ &=\frac{i}{2a}\,\oint_C \frac{z^n}{(z-a)(z-1/a)}dz+\frac{i}{2a}\,\oint_C \frac{z^{-n}}{(z-a)(z-1/a)}dz \end{align}$$

Note that the first integral has a singularity at $z=a$ only (since $|a|<1$; we examine the case $|a|>1$ in Note 1), while integrand has singularities at both $z=0$ (a pole of order $n$) and $z=a$. To evaluate the first integral one needs to determine the residue at $a$. This is simply, $\frac{a^n}{a-1/a}=\frac{a^{n+1}}{a^2-1}$.

In Note 2, we will exploit symmetry to simplify the second integral. This development is different from those shown in other posts inasmuch as the symmetry is exploited after transforming the original integral to a contour integral in the complex plane.

Here, we proceed "brute force." We need to determine the residues at both $0$ and $a$. The residue at $a$ is straightforward and is $a^{-n}/(a-1/a)=a^{-n+1}/(a^2-1)$. For the residue at $0$, we use the formula for an $n$th order pole

$$\begin{align} \text{Res}_{z=0}&=\frac{1}{(n-1)!}\lim_{z\to 0} \frac{d^{n-1}}{dz^{n-1}}\left(\frac{1}{(z-a)(z-1/a)}\right)\\\\ &=\frac{1}{(n-1)!}\lim_{z\to 0} \sum_{k=0}^{n-1} \binom{n-1}{k} \frac{d^{k}}{dz^{k}}\left(\frac{1}{(z-a)}\right)\,\frac{d^{n-1-k}}{dz^{n-1-k}}\left(\frac{1}{(z-1/a)}\right)\\\\ &=a^{-n+1}\sum_{k=0}^{n-1} a^{2k}\\\\ &=\frac{a^{n+1}-a^{-n+1}}{a^2-1} \end{align}$$

Adding the two residues reveals that the second integral is $a^{n+1}/(a^2-1)$, which is identical to the value of the first integral.

Finally, adding the results and multiplying by $(2\pi i) (i/2a)$ yields

$$\int_{-\pi}^{\pi}\frac{\cos (n\theta)}{1-2a\cos \theta+a^2}d\theta=2\pi \frac{a^n}{1-a^2}$$

which agrees with previously obtained results.

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Note 1: Integral for $|a|>1$

Note that the entire development for the case $|a|>1$ is trivial given the symmetry of the integrand in $a$ and $1/a$. Thus we can immediately write

$$\begin{align} \int_{-\pi}^{\pi}\frac{\cos (n\theta)}{1-2a\cos \theta+a^2}d\theta&= 2\pi \frac{a^{-n}}{1-a^{-2}}\\\\ &=2\pi \frac{a^{-n+2}}{a^2-1} \end{align}$$

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Note 2: Exploiting Symmetry

As was shown in other posts, symmetry can be exploited up front in the original integral representation. Here, we show that this tactic can be delayed until after recasting the problem as a contour integral. To that end, we write

$$\begin{align} \int_{-\pi}^{\pi}\frac{\cos (n\theta)}{1-2a\cos \theta+a^2}d\theta&=\frac{i}{2a}\,\oint_C \frac{z^n+z^{-n}}{(z-a)(z-1/a)}dz\\\\ &=\frac{i}{2a}\,\left(\oint_C \frac{z^n}{(z-a)(z-1/a)}dz+\oint_C \frac{z^{-n}}{(z-a)(z-1/a)}dz\right) \end{align}$$

and focus on the second integral on the right-hand side of the last expression. Here, we make the substitution $z=w^{-1}$. Note that (1) $dz=-w^{-2}dw$, (2) the contour's orientation transforms from counter-clockwise to clockwise, (3) $z^{-n}=w^{n}$, and (4) $(z-a)(z-1/a)=w^{-2}(w-a)(w-1/a)$. Implementing the substitution shows that

$$\begin{align} \oint_C \frac{z^{-n}}{(z-a)(z-1/a)}dz=\oint_C \frac{z^{n}}{(z-a)(z-1/a)}dw \end{align}$$

where the contour $C$ is identical (i.e., counter-clockwise) as the minus sign in the differential has been absorbed. Thus, the integral is an even function of $n$ and we need only evaluate the integral with $z^n$ in the numerator.