Contour integration problem with sin and cos: $\int_0^{2\pi}\frac{\sin^2x}{a+b \cos x}dx$

Question

so I'm revising contour integration for an upcoming complex analysis exam. I have been asked to integrate $$\int_0^{2\pi}\frac{\sin^2x}{a+b \cos x}dx$$

I thought the sensible thing to do here would be to substitute in $z=e^{ix}$ and take a contour integral around the unit circle, call this path $\ast$ so that my integral becomes $$\frac{1}{2i}Re(\int_\ast\frac{1-z^2}{az+bz^2}dz)$$

Then, letting $f(z)=\frac{1-z^2}{az+bz^2}$, I thought the function had simple poles at $z=0$ with residue $\frac{1}{a}$ and another simple pole at $z=\frac{-a}{b}$ with residue $\frac{a}{b^2}-\frac{1}{a}$ and thus I get the that $$Re(\int_\ast\frac{1-z^2}{az+bz^2}dz)=2i(2\pi i)\frac{a}{b^2}=-4\pi(\frac{a}{b^2})$$ which is not the answer given, that is: $=\frac{2\pi}{b^2}[a-\sqrt{a^2-b^2}]$,but I can't work out why.

Any help appreciated, thank you in advance.

Answer 1

If you substitute $z = e^{ix}$, then $\sin(x) = (z - \frac{1}{z})/(2i) $ and $\sin^2(x) =(z-\frac{1}{z})^2/(-4) = \frac{(z^2-1)^2}{-4z^2} $

$dz = ie^{ix}dx = izdx$ or $dx = \frac{dz}{zi}$

$a+b\cos x = a + b\frac{z+\frac{1}{z}}{2} = \frac{2az + bz^2 +1}{2z}$

So,

$$I = \int_*\frac{\frac{(z^2-1)^2}{-4z^2}dz}{zi\frac{2az + bz^2 +1}{2z}}$$

$$I = \int_*\frac{i(z^2-1)^2dz}{2z^2(2az + bz^2 +1)}$$

Answer 2

We suppose $b\ne 0$, and get rid of it by force, use rather instead of $a,b$ only the variable $c=a/b$.

(After some edits in the OP we have indeed $0<b<a$, thus $1<c$.)

We will assume $c$ real, $|c|>1$, so that there is no zero in the denominator. (And the integral does not diverge.)

Sooner or later we will have to fight against the two roots of $z^2+2c+1$, we denote them by $U,V$, their product is $UV=1$, and thus we may and do assume $|U|<1$, $|V|>1$.

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Let $C$ be the unit circle centered in the origin of the complex plane, then formally using $z=e^{ix}$, $\frac 1{iz}\; dz=dx$ we have: $$ \begin{aligned} J &= \int_0^{2\pi}\frac{\sin^2x}{a+b\cos x}\;dx \\ &= \frac 1b\int_0^{2\pi}\frac{\sin^2x}{\cos x+c}\;dx \\ &= \frac 1b \int_C \frac{ \left(\frac 1{2i}\left(z-\frac 1z\right)\right)^2} {\frac 12\left(z+\frac 1z\right)+c}\;\frac 1{iz}\;dz \\ &= \frac i{2b} \int_C \frac{ (z^2-1)^2} {z^2(z^2+2cz+1)}\;dz\ . \\[3mm] &\qquad\text{ The partial fraction decomposition is:} \\ \frac{ (z^2-1)^2} {z^2(z^2+2cz+1)} &= \frac{ (z^2-1)^2} {z^2(z-U)(z-V)} \\ &= 1+\frac 1{z^2}+\frac{U+V}z -\frac {U-V}{z-V} -\frac {V-U}{z-U} \ . \\[3mm] &\qquad\text{ Only the residues in $0,U$ contribute, so...} \\ J&=2\pi i\cdot \frac i{2b} \cdot[(U+V)+(U-V)] \\ &=2\pi \frac {-U}b =2\pi \frac {c-\sqrt{c^2-1}}b \ . \end{aligned} $$ The last expression corresponds to the expected answer from the OP, recalling that $c=a/b$.

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Sage check for the partial fraction decomposition:

sage: var('U,z');
sage: V = 1/U
sage: EE = (z^2-1)^2 / z^2 / (z-U) / (z-V)
sage: EE.partial_fraction(z)
-(U^2 - 1)/(U*z - 1) - (U^2 - 1)/((U - z)*U) + (U^2 + 1)/(U*z) + 1/z^2 + 1
sage: bool( _ == (1 + 1/z^2 + (U+V)/z - (V-U)/(z-U) - (U-V)/(z-V) ) )
True

Answer 3

$$ \begin{align} \int_0^{2\pi}\frac{\color{#C00}{\sin^2(x)}}{\color{#090}{a+b\cos(x)}}\,\color{#00F}{\mathrm{d}x} &=\oint\frac{\color{#C00}{-\frac{z^4-2z^2+1}{4z^2}}}{\color{#090}{\frac{bz^2+2az+b}{2z}}}\color{#00F}{\frac{\mathrm{d}z}{iz}}\\ &=-\frac1{2i}\oint\frac{z^4-2z^2+1}{bz^2+2az+b}\,\frac{\mathrm{d}z}{z^2} \end{align} $$ The residue of $\frac{z^4-2z^2+1}{bz^2+2az+b}\frac1{z^2}$ at $z=\frac{-a\pm\sqrt{a^2-b^2}}b$ (simple poles) is $\pm\frac{2\sqrt{a^2-b^2}}{b^2}$ and the residue at $z=0$ (degree $2$ pole) is $-\frac{2a}{b^2}$. Assuming $a\gt0$, we get $2\pi i$ times the sum of the residues inside the unit circle, $\frac{-a+\sqrt{a^2-b^2}}b$ and $0$, to be $$ \int_0^{2\pi}\frac{\sin^2(x)}{a+b\cos(x)}\,\mathrm{d}x=2\pi\frac{a-\sqrt{a^2-b^2}}{b^2} $$