Why $ \int_0^{2\pi}\frac{\cos t - r}{1 - 2r\cos t + r^2}\,dt=0$ for $r\in[0,1)$

Question

Define $$ I(r) = \int_0^{2\pi}\frac{\cos t- r}{1 - 2r\cos t + r^2}\,dt $$ over $r\in [0,1)$. Numerical experiments suggest $I(r) = 0$ for all $r\in [0,1)$. But I can't show this analytically.

This integral appears when computing the Cauchy transform of $\overline z$ over a unit circle. The latter, therefore, seems to be constantly zero.

Answer 1

An alternative proof using complex methods:

For $0< r < 1$ let $$ f_r \colon B_\frac{1}{r} (0) \to \mathbb{C} \, , \, f_r(z) = \frac{- \ln(1-rz)}{z} \, , $$ where $f_r(0) = r $ . Then $f_r$ is holomorphic, so $$ I(r) \equiv - \int \limits_0^{2\pi} \ln(1-r \mathrm{e}^{\mathrm{i}t}) \, \mathrm{d} t = - \mathrm{i} \int \limits_{S^1} f_r(z) \, \mathrm{d} z = 0 $$ holds by Cauchy's theorem. If you are not familiar with complex analysis, you can also show this using the Taylor series of the logarithm: $$ I(r) = \sum \limits_{n=1}^\infty \frac{r^n}{n} \int \limits_0^{2\pi} \mathrm{e}^{\mathrm{i} n t} \, \mathrm{d} t = 0 \, . $$ This implies \begin{align} \int \limits_0^{2\pi} \frac{\cos(t) - r}{1 - 2 r \cos(t) + r^2} \, \mathrm{d} t &= - \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d} r} \int \limits_0^{2\pi} \ln(1 - 2 r \cos(t) + r^2) \, \mathrm{d} t \\ &= - \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d} r} \int \limits_0^{2\pi} \ln[(1 -r \mathrm{e}^{\mathrm{i} t})(1 -r \mathrm{e}^{-\mathrm{i} t})] \, \mathrm{d} t \\ &= \frac{\mathrm{d}}{\mathrm{d} r} I(r) = \frac{\mathrm{d}}{\mathrm{d} r} 0 = 0 \end{align} as desired.

Answer 2

Hint: $$\cos(t)=\frac{1-a^2}{1+a^2}$$ and $$dt=\frac{2da}{1+a^2}$$ Now your indefinite integral is given by $$\int-\frac{2 \left(a^2 r+a^2+r-1\right)}{\left(a^2+1\right) \left(a^2 r^2+2 a^2 r+a^2+r^2-2 r+1\right)}da$$ and this is rational The solution of this integral is given by $$-2 \left(\frac{\tan ^{-1}(a)}{2 r}-\frac{\tan ^{-1}\left(\frac{a (r+1)}{1-r}\right)}{4 r}+\frac{\tan ^{-1}\left(\frac{a (r+1)}{r-1}\right)}{4 r}\right)$$

Answer 3

(Since it is not mentioned yet) The function $$ P_r(t) = \sum_{n=-\infty}^\infty r^{|n|} e^{int} =\frac{1-r^2}{1-2r\cos t +t^2},\quad0\le r<1, 0\le t\le 2\pi $$ is called the Poisson kernel. For any continuous function $f:[0,2\pi]\to \Bbb C$ the Poisson integral $$ u(r,\theta) = \frac{1}{2\pi i}\int_0^{2\pi} f(t)P_r(\theta-t)\mathrm{d}t\tag{*} $$ gives the unique solution of the Dirichlet problem $\triangle u =0, \lim\limits_{r\to 1^-}u(r,\theta)=f(\theta)$.

Let $u(r,\theta)=r\cos \theta$. Since it is a real part of the analytic function $z=r e^{i\theta}$, we find that $$\triangle u=0,\ \ \ \lim\limits_{r\to 1^-}u(r,\theta)=\cos \theta.$$ By $\text{(*)}$ it follows that $$ r\cos \theta =\frac{1}{2\pi}\int_0^{2\pi} \frac{\cos t(1-r^2)}{1-2r\cos (t-\theta)+r^2}\mathrm{d}t $$ for all $0\le r<1$ and $0\le \theta\le 2\pi$. Using the fact $$ \frac{1}{2\pi}\int_0^{2\pi} \frac{1-r^2}{1-2r\cos (t-\theta)+r^2}\mathrm{d}t=\frac{1}{2\pi}\sum_{n=-\infty}^\infty r^{|n|} \int_0^{2\pi}e^{in(t-\theta)}\mathrm{d}t =1, $$ it follows $$ \frac{1}{2\pi}\int_0^{2\pi} \frac{\left(\cos t-r\cos \theta\right)(1-r^2)}{1-2r\cos (t-\theta)+r^2}\mathrm{d}t =0. $$ By letting $\theta =0$, we obtain $$ \frac{1-r^2}{2\pi }\int_0^{2\pi} \frac{\cos t-r}{1-2r\cos t+r^2}\mathrm{d}t =0,$$ or equivalently $$ \int_0^{2\pi} \frac{\cos t-r}{1-2r\cos t+r^2}\mathrm{d}t =0.$$

Answer 4

It seems to me that this result is a particular case (or rather 1D adaptation) of Newton's shell theorem. In 2D, the theorem states that no net gravitational force is perceived by objects located inside a uniform hollow sphere.

Here, consider the integral $$ \vec I(M) = \int_{X\in C}\frac{\vec{MX}}{MX^2}\mathrm ds $$ where $M$ is a point at a distance $r<1$ away from the center $O$ of the unit circle $C$, and $\mathrm ds$ is a length element of the circle. What you are asking for is a component of $\vec I$, the one directed along $\vec{OM}$. In fact, we have $\vec I=\vec 0$. There is a nice geometric proof of this in 2D due to Newton; see here. You can try to adapt it to 1D.

Answer 5

Here is yet another slightly different approach.

Let $$P_r(t) = \frac{\cos t - r}{1 - 2r \cos t + r^2}, \qquad r \in [0,1).$$ Using $\cos t = (e^{it} + e^{-it})/2$, we can rewrite $P_r(t)$ as \begin{align} P_r (t) &= \frac{1}{2} \frac{(e^{it} + e^{-it}) - 2r}{1 - r(e^{it} + e^{-it}) + r^2}\\ &= -\frac{1}{2} \frac{2r - e^{it} - e^{-it}}{(r - e^{it})(r - e^{-it})}\\ &= -\frac{1}{2} \left [\frac{1}{r - e^{it}} + \frac{1}{r - e^{-it}} \right ]\\ &= \frac{1}{2} \left [\frac{e^{-it}}{1 - r e^{-it}} + \frac{e^{it}}{1 - r e^{it}} \right ]\\ &= \frac{1}{2} \left [e^{-it} \sum_{n = 0}^\infty r^n e^{-int} + e^{it} \sum_{n = 0}^\infty r^n e^{int} \right ]\\ &= \sum_{n = 0}^\infty r^n \left (\frac{e^{i(n + 1)t} + e^{-i(n + 1)t}}{2} \right )\\ &= \sum_{n = 0}^\infty r^n \cos (n + 1)t. \end{align}

Now \begin{align} \int^{2\pi}_0 P_r (t) \, dt = \int^{2 \pi}_0 \sum_{n = 0}^\infty r^n \cos (n + 1)t \, dt = \sum_{n = 0}^\infty r^n \int^{2\pi}_0 \cos (n + 1)t \, dt = 0. \end{align}

Answer 6

With $\gamma$ being the counter-clockwise unit circle and $\bar\gamma$ being the clockwise unit circle, $$ \begin{align} \int_0^{2\pi}\frac{r-\cos(t)}{1-2r\cos(t)+r^2}\,\mathrm{d}t &=\int_0^{2\pi}\frac12\left(\frac1{r-e^{it}}+\frac1{r-e^{-it}}\right)\mathrm{d}t\tag1\\ &=\frac1{2i}\int_\gamma\frac{\,\mathrm{d}z}{z(r-z)}-\frac1{2i}\int_{\bar\gamma}\frac{\,\mathrm{d}z}{z(r-z)}\tag2\\ &=\frac1i\int_\gamma\frac{\,\mathrm{d}z}{z(r-z)}\tag3\\ &=\frac1{ir}\int_\gamma\left(\frac1z-\frac1{z-r}\right)\mathrm{d}z\tag4\\ &=\left\{\begin{array}{} 0&\text{if }|r|\lt1\\ \frac{2\pi}r&\text{if }|r|\gt1 \end{array}\right.\tag5 \end{align} $$ Explanation:
$(1)$: partial fractions
$(2)$: $z=e^{it}$ in the left integral and $z=e^{-it}$ in the right integral
$(3)$: $\bar\gamma$ is in the opposite direction from $\gamma$
$(4)$: partial fractions
$(5)$: $2\pi i$ times the sum of the residues inside $\gamma$

Answer 7

Note that $$I(r) =\frac{-1}{2r}\int_{0}^{2\pi}\frac{1-2r\cos t+r^2+r^2-1}{1-2r\cos t+r^2}\,dt$$ which is same as $$-\frac{\pi} {r} +\frac{1-r^2}{r}\int_{0}^{\pi}\frac{dt}{1+r^2-2r\cos t} $$ The integral above is equal to $$\frac{\pi} {\sqrt{(1+r^2)^2-4r^2}}=\frac{\pi}{1-r^2} $$ and hence $I(r) =0$.

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We have used the formula $$\int_{0}^{\pi}\frac{dt}{a+b\cos t} =\frac{\pi} {\sqrt{a^2-b^2}},\,a>|b|$$ which can be proved using the substitution $$(a+b\cos t) (a-b\cos u) =a^2-b^2$$

Answer 8

Not sure why none of the answers mention this approach, but you can just take the derivative with respect to $r$:

\begin{align} I'(r) &= \int_0^{2\pi}\frac{d}{dr}\left(\frac{\cos t - r}{1 - 2r\cos t + r^2}\right)\,dt\\ &= \int_0^{2\pi} \frac{(r^2-1)-2r\cos t + 2\cos^2t}{(1 - 2r\cos t + r^2)^2} \\ &= \frac1{i}\int_{|z|=1} \frac{(r^2-1)-r(z+z^{-1}) + \frac12(z+z^{-1})^2}{(1 - r(z+z^{-1}) + r^2)^2}\cdot\frac{dz}{z}\\ &= \frac1{i}\int_{|z|=1} \frac{z^4-2r z^3+2r^2z^2-2rz+1}{(r^2+1)z(z-r)^2\left(z-\frac1r\right)^2}\,dz\\ \end{align}

Since $r \in [0,1\rangle$, the relevant residues are $$\operatorname{Res}(f,0) = \frac1{2r^2}$$ $$\operatorname{Res}(f,r) = -\frac1{2r^2}$$ so $I'(r) = 0$ for $r \in [0,1\rangle$.

We conclude that $I$ is constant on $[0,1\rangle$. Hence $$I(r)=I(0) = \int_0^{2\pi}\cos t\,dt = 0$$

Answer 9

From the identity

$$\sum_{n=1}^{\infty}r^{n-1} \cos(nt)=\frac{\cos(t)-r}{1-2r\cos(t)+r^2}, \ |r|<1$$

it follows that

$$I(r)=\sum_{n=1}^\infty r^{n-1}\int_0^{2\pi}\cos(nt)dt=\sum_{n=1}^\infty\frac{r^{n-1}\sin(2n\pi)}{n}=0$$

Answer 10

$$ \int_0^{2\pi}\frac{\cos t - r}{1 - 2r\cos t + r^2}\,dt =\frac1r \cot^{-1}\left(\frac1r\csc t -\cot t\right)\bigg|_0^{2\pi}=0 $$