How can we show that $\int_{0}^{2\pi}{x\over \phi-\cos^2(x)}\mathrm dx=2\pi^2?$

We have the integral

$$\int_{0}^{2\pi}{x\over \phi-\cos^2(x)}\mathrm dx=2\pi^2\tag1$$ $\phi$; Golden ratio

What method do we employ to prove $(1)$?

An attempt:

If we use $u=\phi-\cos^2 x$ then $\int_{\phi-1}^{\phi-1}(...)du=0$

Another way, $(1)$ becomes

$${1\over 2\sqrt{\phi}}\int_{0}^{2\pi}\left({x\over \sqrt{\phi}-\cos x}-{x\over \sqrt{\phi}+\cos x} \right)\mathrm dx\tag2$$

We got a hint by using

$z=\tan{x\over 2}$, $dx={2dz\over 1+z^2}$, $\cos x={1-z^2\over 1+z^2}$ and $\sin x={2z\over 1+z^2}$

Let try on

$$\int_{0}^{2\pi}{x\over \sqrt{\phi}+\cos x}\mathrm dx=4\int_{0}^{0}{\tan^{-1} z\over (1+z^2)\sqrt{\phi}+1-z^2}\cdot (1-z^2)\mathrm dz=0?\tag3$$

Doesn't seem to work.

How else can we prove $(1)$?

Solution 1:

Here is a route, I leave it to you to fill in the details: $$ \begin{aligned} \int_0^{2\pi}\frac{x}{\phi-\cos^2x}\,dx &=\int_0^\pi\frac{x}{\phi-\cos^2x}\,dx+\int_0^{\pi}\frac{x+\pi}{\phi-\cos^2(x+\pi)}\,dx\\ &=2\int_0^\pi\frac{x}{\phi+\sin^2x-1}\,dx+\int_0^\pi\frac{\pi}{\phi-\cos^2x}\,dx\\ &=2\frac\pi2\int_0^\pi\frac{1}{\phi+\sin^2x-1}\,dx+\int_0^\pi\frac{\pi}{\phi-\cos^2x}\,dx\\ &=2\pi\int_0^\pi\frac{1}{\phi-\cos^2x}\,dx\\ &=4\pi\int_0^{\pi/2}\frac{1}{\phi-\cos^2x}\,dx\\ &=4\pi\int_0^{\pi/2}\frac{1}{\cos^2x}\frac{1}{\phi\tan^2x+\phi-1}\,dx\\ &=4\pi\biggl[\frac{1}{\sqrt{\phi(\phi-1)}}\arctan\Bigl(\sqrt{\frac{\phi}{\phi-1}}\tan x\Bigr)\biggr]_0^{\pi/2}\\ &=4\pi\frac{1}{\sqrt{\phi(\phi-1)}}\frac\pi2\\ &=2\pi^2. \end{aligned} $$

Solution 2:

Since it may be in any case instructive, let me try to give the full residue-method solution. Choosing the branch cut of the logarithm along the positive real axis, let us apply the residue theorem to the corresponding keyhole contour $\Gamma$, which, as sketched in the figure, goes around the point $z=z_1=\sqrt{2\phi-3}=-z_2$ from above and below. We obtain: $$ \oint_\Gamma \frac{4z\log z}{z^4-2(2\phi-1)z^2+1}dz = i2\pi\frac{\log z_2}{z_2^2-(2\phi-1)} = -i\pi\log z_1 +\pi^2, $$ where we have used that $z_{2}$ is the only root of the denominator enclosed in the integration contour and we have substituted $\log z_2 = \log(z_1e^{i\pi})=\log z_1+i\pi$.

The integration contour $\Gamma$ can be split into its six pieces: the unit circle $C_1$, two segments above and below the real axis, two half-circles $\gamma^\pm_\varepsilon$ around $z_1$ and a small circle around the origin. As the radius $\varepsilon$ of the small circle and of $\gamma_\varepsilon^\pm$ tends to zero, these pieces yield (care is needed because we are comparing quantities across a branch cut): $$\begin{aligned} \oint_\Gamma \frac{4z\log z}{z^4-2(2\phi-1)z^2+1}dz =&\ \int_0^{2\pi}\frac{t}{\phi-\cos^2t}dt\\ &- \mathrm{PV} \int_0^1 \frac{4x(\log x +i2\pi)}{x^4-2(2\phi-1)x^2+1}dx\\ &-i\pi \frac{\log z_1 + i2\pi}{z_1^2-(2\phi-1)}\\ &+0\\ &+\mathrm{PV} \int_0^1 \frac{4x\log x }{x^4-2(2\phi-1)x^2+1}dx\\ &-i\pi \frac{\log z_1}{z_1^2-(2\phi-1)}\\ =&\int_0^{2\pi}\frac{t}{\phi-\cos^2t}dt -\pi^2\\ &-i8\pi\, \mathrm{PV}\int_0^1\frac{x}{x^4-2(2\phi-1)x^2+1}dx+i\pi\log z_1, \end{aligned} $$ where $\mathrm{PV}$ denotes the Cauchy principal value. By comparison of the real parts, $$ \int_{0}^{2\pi}\frac{t}{\phi-\cos^2t}dt=2\pi^2. $$ Furthermore, by comparison of the imaginary parts, we get the bonus identity $$ \mathrm{PV} \int_{0}^{1}\frac{ x}{x^4-2(2\phi-1)x^2+1}dx=\frac{1}{4}\log z_1=\frac{1}{8}\log(\sqrt{5}-2). $$ Alio modo: let us consider instead the complex function $$ f(z)=\frac{z}{\phi-\cos^2z}=\frac{-4ze^{i2z}}{(e^{i2z}-e^{i2\zeta_1})(e^{i2z}-e^{i2\zeta_3})}, $$ where $\zeta_1=-i\log\sqrt{2\phi-3}=\zeta_2-\pi$ and $\zeta_3=-i\log\sqrt{2\phi+1}=\zeta_4-\pi$. This function has simple poles at $\zeta_1$, $\zeta_2$ in the upper half plane and $\zeta_3$, $\zeta_4$ in the lower half plane, up to integer multiples of $2\pi$. Denoting by $\Pi$ the rectangular contour sketched in the figure , with half-circular indents of radius $\varepsilon$ around $\zeta_1$ and $\zeta_1+2\pi$, we have $$ \oint_\Pi f(z) dz=i2\pi \,\text{Res}f(\zeta_2)=i2\pi\lim_{z\to\zeta_2}(z-\zeta_2)f(z)=\pi^2-i\pi\log\sqrt{2\phi-3}. $$ Furthermore, in the limits $M\to\infty$ and $\varepsilon\to0$, from the various pieces of the integration contour we get: $$\begin{aligned} \oint_\Pi f(z) dz =&\ \int_0^{2\pi}\frac{t}{\phi-\cos^2t}dt\\ &\ +\mathrm{PV} \int_0^\infty \frac{2\pi+iy}{\phi-\cosh^2y}idy\\ &\ -0\\ &\ -\mathrm{PV} \int_0^\infty \frac{iy}{\phi-\cosh^2y}idy\\ &\ -i\pi \,\text{Res}f(\zeta_1)\\ &\ -i\pi \,\text{Res}f(\zeta_1+2\pi)\\ =&\ \int_0^{2\pi}\frac{t}{\phi-\cos^2t}dt-\pi^2\\ &\ i2\pi\, \mathrm{PV} \int_0^\infty \frac{dy}{\phi-\cosh^2y}+i\pi\log\sqrt{2\phi-3}. \end{aligned}$$ Again, by comparison: $$ \begin{aligned} \int_0^{2\pi}\frac{t}{\phi-\cos^2t}dt&=2\pi^2\\ \mathrm{PV} \int_0^\infty \frac{dy}{\phi-\cosh^2y}&=\frac{1}{2}\log(2\phi-3). \end{aligned} $$

Solution 3:

Instead of starting from your integral, I will start from the integral in the solution by @Mickep, i.e,

$$\tag{1}I=4\pi\int_0^{\pi/2}\frac{1}{\phi-\cos^2x}\,dx=\pi\underbrace{\int_0^{2\pi}\frac{1}{\phi-\cos^2x}\,dx}_{J} $$

on which it will be simpler to explain how to compute an integral using residue calculus than on the original integral. I will not enter into details. For this I refer you to the numerous lecture notes on residue calculus, for example here.

The main idea is to transform this integral into a circuit integral along a closed parametrized arc, which is here, in a natural way, the unit circle $\gamma$, traversed in the direct orientation.

Let $z=e^{ix}$, with $dz=ie^{ix}dx$. Thus, $J$ becomes, using Euler formula $\cos(x)=\frac12(e^{ix}+e^{-ix})$: $$J=\int_{\gamma}\dfrac{-i dz/z}{\Phi-\dfrac{(z+1/z)^2}{4}}$$

Expanding and reducing, we get:

$$\tag{2}J=-i \underbrace{\int_{\gamma}\dfrac{4z dz}{-z^4+2(2\Phi-1)z^2-1}}_{K}$$

The integrand has four poles (roots of the denominator), all of them real:

$$\begin{cases}z_1=\sqrt{2 \Phi -3} \ \ \ \text{and } \ \ \ z_2=-z_1=-\sqrt{2 \Phi -3}\\ z_3=\sqrt{1+2 \Phi} \ \ \ \text{and } \ \ \ z_4=-z_3=-\sqrt{1+2 \Phi}\end{cases}$$

Only $z_1$ and $z_2$ are inside contour $\gamma.$

The residue theorem says that a complex integral $\int_{\gamma}...$ whatever the sufficiently regular closed contour $\gamma$, is equal to the sum of residues at the poles situated inside the contour multiplied by $2i\pi$.

Now, what is the residue of a function of the form $\dfrac{f(z)}{g(z)}$ at a pole $z_0$? It is the number given by the following formula:

$$\dfrac{f(z_0)}{g'(z_0)}.$$ Although this is not the most general definition, this expression covers a large number of cases (see remark below).

Thus, the residue theorem gives:

$$K=2i\pi\left(\dfrac{4z_1 }{-4z_1^3+4(2\Phi-1)z_1}+\dfrac{4z_2}{-4z_2^3+4(2\Phi-1)z_2}\right)$$

$$K=2i\pi\left(\dfrac{1}{-z_1^2+(2\Phi-1)}+\dfrac{1}{-z_2^2+(2\Phi-1)}\right)$$

As $z_1^2=z_2^2=2\Phi-3$, the denominators have a common value, which is $-2$. Thus $K=-2i\pi$. Plugging this value into (2) and then (1) gives the awaited result.

Remark: The definition of residues I have given is only valid for simple poles; if $z_0$ is for example a double root of the denominator, it is understandable that we are in trouble because $g'(z_0)=0$. There exist specific formulas for these cases.

Solution 4:

A purely real approach.

Consider the integral $$C(a,b)=\int_0^{\pi}\frac{\mathrm dx}{a+b\cos(x)^2}$$ It has already been shown by other users that your integral is given by $2\pi C(\phi,-1)$.

We commence with the substitution $t=\tan\frac{x}2$: $$C(a,b)=2\int_0^\infty \frac1{a+b\left[\frac{1-t^2}{1+t^2}\right]^2}\frac{\mathrm dt}{1+t^2}$$ $$C(a,b)=2\int_0^\infty\frac{1+t^2}{a(t^2+1)^2+b(t^2-1)^2}\mathrm dt$$ $$C(a,b)=2\int_0^\infty\frac{1+t^2}{(a+b)t^4+2(a-b)t^2+a+b}\mathrm dt$$ $$C(a,b)=\frac2{a+b}\int_0^\infty\frac{1+t^2}{t^4+2gt^2+1}\mathrm dt$$ Where $g=\frac{a-b}{a+b}$. We may split up the integral: $$C(a,b)=\frac2{a+b}\int_0^\infty\frac{\mathrm dt}{t^4+2gt^2+1}+\frac2{a+b}\int_0^\infty\frac{t^2\mathrm dt}{t^4+2gt^2+1}$$ Then we pay attention to $$N_{n}(s)=\int_0^\infty \frac{x^{2n}}{x^4+2sx^2+1}\mathrm dx$$ With the substitution $x=1/u$, we have $$N_n(s)=-\int_\infty^0 \frac{\frac1{u^{2n}}}{\frac1{u^4}+\frac{2s}{u^2}+1}\frac{\mathrm du}{u^2}$$ $$N_n(s)=\int_0^\infty \frac{u^{2-2n}\mathrm du}{u^4+2su^2+1}$$ $$N_n(s)=N_{1-n}(s)$$ So we have $$C(a,b)=\frac4{a+b}\int_0^\infty\frac{\mathrm dt}{t^4+2gt^2+1}$$ For this remaining integral, we recall that $$\begin{align} \frac1{x^4+(2-c^2)x^2+1}&=\frac1{(x^2+cx+1)(x^2-cx+1)}\\ &=\frac1{4c}\frac{2x+c}{x^2+cx+1}-\frac1{4c}\frac{2x-c}{x^2-cx+1}+\frac14\frac1{x^2+cx+1}+\frac14\frac1{x^2-cx+1} \end{align}$$ So we have that $$K(c)=\int_0^\infty \frac{\mathrm dx}{x^4+(2-c^2)x^2+1}$$ becomes $$K(c)=\frac1{4c}\int_0^\infty\frac{2x+c}{x^2+cx+1}\mathrm dx-\frac1{4c}\int_0^\infty\frac{2x-c}{x^2-cx+1}\mathrm dx\\ +\frac14\int_0^\infty \frac{\mathrm dx}{x^2+cx+1}+\frac14\int_0^\infty \frac{\mathrm dx}{x^2-cx+1}$$ The first two integrals are easy, and end up vanishing, so we end up with $$K(c)=\frac14I(1,c,1)+\frac14I(1,-c,1)$$ Where $$I(a,b,c)=\int_0^\infty \frac{\mathrm dx}{ax^2+bx+c}=\frac2{\sqrt{4ac-b^2}}\arctan\frac{2ax+b}{\sqrt{4ac-b^2}}\bigg]_0^\infty\\ =\frac\pi{\sqrt{4ac-b^2}}-\frac2{\sqrt{4ac-b^2}}\arctan\frac{b}{\sqrt{4ac-b^2}}$$ Hence $$K(c)=\frac\pi{2\sqrt{4-c^2}}$$ So we see that $$C(a,b)=\frac4{a+b}K\left(2\sqrt{\frac{b}{a+b}}\right)=\frac\pi{\sqrt{a^2+ab}}$$ And since, by definition, $\phi^2-\phi=1$, we have that $$C(\phi,-1)=\pi$$ Which gives, as desired, $$\int_0^{2\pi}\frac{x\mathrm dx}{\phi-\cos(x)^2}=2\pi^2$$