# Show that $\int_0^{2\pi}\frac{R^2-r^2}{R^2 - 2Rr\cos (\varphi-\vartheta) + r^2}d\vartheta$ is independent of $R>r>0$, using only real numbers.

The poisson kernel is sometimes written as $$\frac{1}{2\pi}\int_0^{2\pi} \frac{R^2-r^2}{R^2 - 2Rr\cos(\varphi-\vartheta) + r^2} \mathrm{d}\vartheta = 1 \ , \ \ R>r>0$$ Where $\varphi$ is some arbitary angle. This much used in complex analysis amongst other fields. Is there some basic, elementary way of showing that the integral is independent on $R$ and $r$?

Splitting the integral at $\int_0^\pi + \int_\pi^{2\pi}$ and using the Weierstrass substitution seems like somewhat ugly and a messy way to approach the problem.

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Maybe a stupid question: it is possible to differentiate with respect to $R$ and $r$? I never tried to. –  Siminore Jan 25 at 12:59
Is $d\theta$ at the right place ? Please correct also in the title. –  Claude Leibovici Jan 25 at 13:08
If you want solution using real methods, then I don't understand (complex-analysis) tag. –  user110822 Jan 25 at 13:42

Step I. Using integration by parts one can easily show that $$J_n=\int_0^{2\pi} \cos^{2n}x\,dx=\frac{2\pi}{4^n}\binom{2n}{n},$$ as for the cosine integral we make use of a recurrence, \begin{align} J_{n+1}&=\int_0^{2\pi} \cos^{2(n+1)}x\,dx=\int_0^{2\pi} (\cos^{2n+1}x)(\sin x)'\,dx =0+(2n+1)\int_0^{2\pi} \cos^{2n}x\,\sin^2 x\,dx \\&= (2n+1)\int_0^{2\pi} \cos^{2n}x\,(1-\cos^2 x)\,dx=(2n+1)J_n-(2n+1)J_{n+1}, \end{align} and thus $J_{n+1}=\frac{2n+1}{2n+2}J_n=2^{-n-1}\prod_{j=0}^{n+1}\frac{2j-1}{j+1}J_0=\frac{2\pi}{4^{n+1}}\frac{(2n+2)!}{(n+1)!(n+1)!}$.

Step II. If $|a|<1$, then $$\frac{1}{\sqrt{1-a^2}}=(1-a^2)^{-1/2}=\sum_{n=0}^\infty \binom{-\frac{1}{2}}{n}a^{2n}=\cdots=\sum_{n=0}^\infty \binom{2n}{n}\frac{a^{2n}}{2^{2n}}.$$

Step III. Then $$I=\frac{1}{2\pi}\int_0^{2\pi} \frac{R^2-r^2}{R^2 - 2Rr\cos(\phi-\theta) + r^2} \mathrm{d}\theta =\frac{R^2-r^2}{2\pi(R^2+r^2)}\int_0^{2\pi}\frac{dx}{1-a\cos x},$$ where $a=\dfrac{2rR}{r^2+R^2}.$ Also $$\int_0^{2\pi}\frac{dx}{1-a\cos x}=\sum_{n=0}^\infty\int_0^{2\pi} a^{2n}\cos^{2n}x\,dx=\sum_{n=0}^\infty \frac{2\pi\,a^{2n}}{4^n}\binom{2n}{n}=\frac{2\pi}{\sqrt{1-a^2}}$$

Step IV. Finally, \begin{align} I&=\frac{1}{2\pi}\int_0^{2\pi} \frac{R^2-r^2}{R^2 - 2Rr\cos(\phi-\theta) + r^2} \mathrm{d}\theta =\frac{R^2-r^2}{2\pi(R^2+r^2)}\int_0^{2\pi}\frac{dx}{1-a\cos x} \\ &=\frac{R^2-r^2}{2\pi(R^2+r^2)}\cdot \frac{2\pi}{\sqrt{1-\left(\frac{2rR}{r^2+R^2}\right)^2}} =\cdots=1. \end{align}

Note. I prefer the proof of T.A.E, but the question was asking of a proof without the use of complex numbers!

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Because you're integrating over a full period of $\cos$ and because $R > 0$, your expression becomes $$\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1-(r/R)^{2}}{1-2(r/R)\cos\theta+(r/R)^{2}}\,d\theta.$$ Let $\rho = r/R$. Then, by assumption, $0 < \rho < 1$, and \begin{align} \frac{1-(r/R)^{2}}{1-2(r/R)\cos\theta+(r/R)^{2}} & =\frac{1-\rho^{2}}{(1-\rho e^{i\theta})(1-\rho e^{-i\theta})} \\ & = \frac{1}{1-\rho e^{i\theta}}+\frac{\rho e^{-i\theta}}{1-\rho e^{-i\theta}} \\ & = \sum_{n=-\infty}^{\infty}\rho^{|n|} e^{in\theta} \\ & = 1 + 2\sum_{n=1}^{\infty}\rho^{n}\cos(n\theta). \end{align} When you integrate the terms in this sum over $0 \le \theta \le 2\pi$, only the integral of the constant term is non-zero. The final integral value is 1, regardless of $0 \le \rho < 1$.

If you don't like the complex exponential in between, just verify directly that $$F(\rho) = \frac{1}{2\pi}\int_{0}^{2\pi}\frac{1-\rho^{2}}{1-2\rho\cos\theta+\rho^{2}}d\theta$$ satisfies the differential equation $$F''(\rho)+\frac{1}{\rho}F'(\rho)=0.$$ This a separable equation whose solutions have the form $$F'(\rho) = C\frac{1}{\rho} \\ F(\rho) = C\ln \rho + D.$$ The constant $C$ must be $0$ in this case for obvious reasons.

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You know i explicitly asked for solutions using real methods? =) Still a nice answer, so just let it be, but unfortunately you will probably not get the accept. –  N3buchadnezzar Jan 25 at 13:34
@N3buchadnezzar: I have added the solution for you that depends only on real variables. –  T.A.E. Jan 25 at 14:10


The integral is obviously independent of $\phi$: Just derive the integral respect of $\phi$ and use $\ds{\partiald{}{\phi} = - \partiald{}{\theta}}$. Since the integrand is a periodic function of $\theta$ it turns out that the $\phi$-derivative of the integral is zero. Then, we just need to consider the integral: $${\cal I} \equiv {1 \over \pi}\int_{0}^{\pi}{1 - h^{2} \over 1 - 2h\cos\pars{\theta} + h^{2}}\,\dd\theta\,,\qquad 0< h \equiv {r \over R} < 1$$

With Weierstrass substitution $\ds{t \equiv \tan\pars{\theta \over 2}}$: \begin{align} {\cal I} &= {1 - h^{2} \over \pi}\int_{0}^{\infty} {1 \over 1 - 2h\pars{1 - t^{2}}/\pars{1 + t^{2}} + h^{2}}\,{2\,\dd t \over 1 + t^{2}} \\[3mm]&= 2\,{1 - h^{2} \over \pi}\int_{0}^{\infty} {\dd t \over \pars{h^{2} + 2h + 1}t^{2} + h^{2} - 2h + 1} = {2 \over \pi}\,\int_{0}^{\infty} {\bracks{\pars{1 + h}/\pars{1 - h}}\,\dd t \over \bracks{\pars{1 + h}t/\pars{1 - h}}^{2} + 1} \\[3mm]&= {2 \over \pi}\ \overbrace{\int_{0}^{\infty}{1 \over t^{2} + 1}}^{\ds{=\ {\pi/2}}} =\color{#00f}{\Large 1} \end{align}

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