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Define a function $f(\alpha, \beta)$, $\alpha \in (-1,1)$, $\beta \in (-1,1)$ as

$$ f(\alpha, \beta) = \int_0^{\infty} dx \: \frac{x^{\alpha}}{1+2 x \cos{(\pi \beta)} + x^2}$$

You can use, for example, the Residue Theorem to show that

$$ f(\alpha, \beta) = \frac{\pi \sin{\pi \alpha \beta}}{ \sin{\pi \alpha} \sin{\pi \beta}} $$

Clearly, from this latter expression, $f(\alpha, \beta) = f(\beta, \alpha)$. My question is, can one see this symmetry directly from the integral expression?

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Fascinating question. –  user7530 Jan 1 '13 at 22:30
I'm tempted to say "no". Seems like the simples way to see this symmetry is to evaluate the integral... –  Fabian Jan 1 '13 at 22:33
Invariance under $\alpha \to -\alpha$ is not obvious either. –  Maesumi Jan 1 '13 at 23:16
@user7530: thanks. This question has vexed me for some time. You'd think that symmetry like this has a simple explanation. But I wonder what I am missing. –  Ron Gordon Jan 2 '13 at 0:57
$f(\alpha, \beta) = \int_{-\infty}^{\infty} du \: \frac{e^{u\alpha}}{e^{-u}+e^u+e^{i \pi \beta}+e^{-i \pi \beta}}$ after $u=\ln x$ and expanding cosine –  valtron Jan 3 '13 at 22:56

4 Answers 4

Very interesting question! But, alas, not an answer. Only few representations for the integral obtained. One of them evaluated to the form claimed in the question.

First, transform the integral into a form, symmetric under $\alpha \mapsto -\alpha$: $$ \int_0^\infty \frac{x^\alpha}{1+2 x \cos(\pi \beta) + x^2} \mathrm{d} x = \int_0^1 \frac{x^\alpha}{1+2 x \cos(\pi \beta) + x^2} \mathrm{d} x + \int_1^\infty \frac{x^\alpha}{1+2 x \cos(\pi \beta) + x^2} \mathrm{d} x $$ Make a change of variables $x \to x^{-1}$ in the last integral to obtain: $$ f(\alpha,\beta) = \int_0^1 \frac{x^\alpha + x^{-\alpha}}{1+2 x \cos(\pi \beta) + x^2} \mathrm{d} x \tag{1} $$ Now, making a change of variables $x = \exp(-t)$ we have: $$ f(\alpha,\beta) = \int_0^\infty \frac{\cosh(\alpha t)}{\cosh(t) + \cos(\beta \pi)} \mathrm{d} t \tag{2} $$ Using $$ \int_0^\infty \exp\left(-u \left( \cosh t + \cos \pi \beta \right) \right) \mathrm{d}u = \frac{1}{\cosh(t) + \cos(\beta \pi)} $$ and the integral representation of the modified Bessel function of the second kind: $$ \int_0^\infty \cosh(\alpha t) \exp\left( - u \cosh t \right) \mathrm{d}t = K_\alpha(u) $$ we arrive at a compact representation: $$ f(\alpha,\beta) = \int_0^\infty K_\alpha(u) \mathrm{e}^{-u \cos\left(\pi \beta\right)} \mathrm{d} u \tag{3} $$ expanding the exponential into series and using $\int_0^\infty u^n K_\alpha(u) \mathrm{d} u = 2^{n-1} \Gamma\left(\frac{n}{2} + \frac{1+\alpha}{2} \right)\Gamma\left(\frac{n}{2} + \frac{1-\alpha}{2} \right)$ we get: $$ f(\alpha,\beta) = \sum_{n=0}^\infty \frac{2^{n-1}}{n!} \left(-\cos \pi \beta\right)^{n} \Gamma\left(\frac{n}{2} + \frac{1+\alpha}{2} \right)\Gamma\left(\frac{n}{2} + \frac{1-\alpha}{2} \right) \tag{4} $$ summing over even and over odd integers: $$ f(\alpha, \beta) = \frac{\pi}{2} \frac{ \cos\left( \alpha \arcsin \cos(\pi \beta) \right) }{ | \sin(\pi \beta) | \cos \left( \frac{\pi \alpha}{2} \right)} - \frac{\pi}{2} \frac{ \sin\left( \alpha \arcsin \cos(\pi \beta) \right) }{ | \sin(\pi \beta) | \sin \left( \frac{\pi \alpha}{2} \right)} = \pi \frac{\sin \left( \alpha \left( \frac{\pi}{2} - \arcsin \cos(\pi \beta) \right) \right)}{ | \sin \pi \beta | \sin(\pi \alpha)} $$ Now $\frac{\pi}{2} - \arcsin \cos(\pi \beta) = \arccos \cos(\pi \beta) = \pi | \beta |$ for $-1<\beta<1$. Thus, restoring parity, we recover the OP's expression: $$ f(\alpha, \beta) = \pi \frac{ \sin(\alpha \beta)}{\sin(\pi \alpha) \sin(\pi \beta)} = \frac{\operatorname{sinc}(\pi \alpha \beta)}{\operatorname{sinc}(\pi \alpha) \operatorname{sinc}(\pi \beta)} \tag{5} $$

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I am very impressed with what you have derived, nonetheless! I love the integral form in Equation (2), and I am out of practice enough to need to review my MacDonald functions to see how you derived that infinite sum, and then evaluated that sum. Nonetheless, as you said, not quite there yet, although I hope that maybe the expression in Equation (2) (user7530's observation taken a few steps further) may help. –  Ron Gordon Jan 4 '13 at 20:19
I am giving you the bounty, although nobody really answered the question. You did provide new insights into this integral for which I am grateful. –  Ron Gordon Jan 10 '13 at 22:01
@rlgordonma You could have awarded the bounty without accepting the answer. Acceptance marks the question with green background that tells other users "nothing to do here, the matter is settled". –  user53153 Jan 11 '13 at 1:23
Ah. Thanks for pointing that out. I will fix that if possible. You should have the bounty nonetheless. –  Ron Gordon Jan 11 '13 at 1:34
@Sasha Exquisite! –  Ahaan Rungta Nov 1 '13 at 17:32

This question is fantastic! I have not found an integral displaying the symmetry but I wanted to show that it points to a symmetry of the Lerch zeta function, if this is known then perhaps that could explain the symmetry, if it is not known it is very interesting I think.

From $(1)$ in @Sasha: $$\begin{aligned} f(\alpha,\beta) = \int_0^1 \frac{x^\alpha + x^{-\alpha}}{1+2 x \cos(\pi \beta) + x^2} \mathrm{d} x \tag{1}\\ \end{aligned} $$ The integrand is comparable to the generating function of the Chebyshev polynomials of the second kind and in fact: $$ \begin{aligned} {\frac {{x}^{\alpha}+{x}^{-\alpha}}{1+2\,x\cos \left( \pi \,\beta \right) +{x}^{2}}}&=\sum _{n=0}^{\infty }U_n \left( - \cos \left( \pi \,\beta \right) \right) \left( {x}^{n+\alpha}+{x}^{n -\alpha} \right) \tag{2}\\ \end{aligned} $$ and the Chebyshev polynomial of the second kind satsifies: $$U_n \left( -\cos \left( \pi \,\beta \right)\right)={\frac { \left( -1 \right) ^{n}\sin \left( \left( 1+n \right) \pi \, \beta \right) }{\sin \left( \pi \,\beta \right) }} \tag{3} $$ so after using $(2,3)$ in $(1)$ and switching integration and summation order we obtain a Fourier series: $$ \begin{aligned} f(\alpha,\beta)&=\frac{1}{\sin \left( \pi \,\beta \right)} \sum _{n=0}^{\infty }\left( -1 \right) ^{n}\sin \left( \left( 1+n \right) \pi \, \beta \right) \left( \dfrac{1}{1+n+\alpha }+ \dfrac{1}{1+n-\alpha } \right)\tag{4}\\ &=-\frac{1}{\sin \left( \pi \,\beta \right)} \sum _{n=-\infty\,(n\ne0)}^{\infty } \dfrac{\left( -1 \right) ^{n}\sin \left( n \pi \, \beta \right)}{n+\alpha } \\ &=-\frac{1}{\sin \left( \pi \,\beta \right)} \mathfrak{I}\left[ \sum _{n=1}^{\infty }\left( -1 \right) ^{n}e^{i n\pi \beta } \left( \dfrac{1}{n+\alpha }+ \dfrac{1}{n-\alpha } \right)\right]\\ &=-\frac{1}{\sin \left( \pi \,\beta \right)} \mathfrak{I}\left[\Phi(-e^{i \pi \beta },1,\alpha)+\Phi(-e^{i \pi \beta },1,-\alpha)\right]\\ &=-\frac{1}{\sin \left( \pi \,\beta \right)} \mathfrak{I}\left[\Phi_{+}(-e^{i \pi \beta },1,\alpha)\right]\quad:\quad(n=-\infty..+\infty,n\ne 0)\\ &=-\frac{1}{\sin \left( \pi \,\beta \right)} \mathfrak{I}\left[L\left(\frac{\beta+1}{2},\alpha,1\right)+L\left(\frac{\beta+1}{2},-\alpha,1\right)\right]\\ &=-\frac{1}{\sin \left( \pi \,\beta \right)} \mathfrak{I}\left[L_{+}\left(\frac{\beta+1}{2},\alpha,1\right)\right]\\ &=-\frac{1}{2\sin \left( \pi \,\beta \right)}\left[L_{+}\left(\frac{\beta+1}{2},\alpha,1\right)-L_{+}\left(\frac{\beta+1}{2},-\alpha,1\right)\right] \end{aligned}$$ where $L$ is the Lerch zeta function and $\Phi$ the Lerch transcendant. I can't believe this Fourier series is invariant under $\alpha \leftrightarrow \beta$. I have not as of yet found that symmetry for the Lerch zeta function on line. This series together with the previous demonstration that: $$f(\alpha, \beta) = \pi \frac{ \sin(\pi\alpha \beta)}{\sin(\pi \alpha) \sin(\pi \beta)} \tag{5}$$ show immediately that the following special cases hold which are interesting in their own right: $$\sum _{n=-\infty }^{\infty }{\frac { \left( -1 \right) ^{n}\sin \left( \pi \,xn \right) }{x-n}}={\frac {\pi\sin \left( \pi{x}^{ 2} \right) }{\sin \left( \pi x \right) }}\tag{6}$$ $$\sum _{n=-\infty }^{\infty }{\frac { \left( -1 \right) ^{n}\cos \left( \pi \,xn \right) }{1- \left( n+x \right) ^{2}}}=\pi\sin \left( \pi {x}^{2} \right)\tag{7}$$

Now, from $(4)$ and $(5)$ and the variable change $\alpha=x,\, \beta=2y-1$, with $-1<x<1,\,0\le y<1$, we have: $$L_{+}(y,x,1)=-L_{+}(y,-x,1)-2\pi i\dfrac{\sin(2\pi x(y-\frac{1}{2})}{\sin(x)} \tag{8}$$ where we recognise the trigonometric term as the Dirichlet Kernel (with $x\rightarrow2x$ and for $y$ generalised to non-integer). If we then use differentiation with respect to $x$ as a raising operator we obtain the reflection formula in the $x$ variable: $$L_{+}(y,x,k)=\left( -1 \right) ^{k-1}L_{+}(y,-x,k) -2\pi i \dfrac{ \left( -1 \right) ^{k-1}}{(k-1)!}{\frac {\partial ^{k-1}}{\partial {x}^{k-1}}} {\frac {\sin \left( 2\pi x \left( y-\frac{1}{2} \right) \right) } {\sin \left( \pi x \right) }} \tag{9}$$ where the order becomes $k$ from simple differentiation of the function definition, and it also follows from reversing summation order in the function definition that: $$L_{+}(y,x,k)=\sum_{n=-\infty (n\ne0)}^{\infty}\dfrac{e^{2\pi in y}}{(n+x)^k}=(-1)^k L_{+}(-y,-x,k)\tag{10}$$ and so $(9)$ can also be viewed as a reflection formula in $y$: $$L_{+}(y,x,k)=L_{+}(-y,x,k) -2\pi i \dfrac{ \left( -1 \right) ^{k-1}}{(k-1)!}{\frac {\partial ^{k-1}}{\partial {x}^{k-1}}} {\frac {\sin \left( 2\pi x \left( y-\frac{1}{2} \right) \right) } {\sin \left( \pi x \right) }} \tag{11}$$ or as an explicit formula for the imaginary part: $$\mathfrak{I}\left(L_{+}(y,x,k)\right)= \pi\dfrac{ \left( -1 \right) ^{k-1}}{(k-1)!}{\frac {\partial ^{k-1}}{\partial {x}^{k-1}}} {\frac {\sin \left( 2\pi x \left( y-\frac{1}{2} \right) \right) } {\sin \left( \pi x \right) }} \tag{11}$$ As a final application, evaluating $(9)$ and $(10)$ at $x=0$ and recognising that $(10)$ is then proportional to the Fourier series of the Bernoulli polynomials $B(m,y)$, we obtain the Taylor series for the Dirichlet kernel in the $x$ variable: $$2\sum _{m=0}^{\infty }\left( -1 \right) ^{m}{\frac { B \left( 2m+1,y \right) }{ \left( 2m+1 \right) !}}{x}^{2m}={ \frac {\sin \left( x \left( y-\frac{1}{2} \right) \right) }{\sin \left( \frac{x}{2} \right) }} \tag{12}$$

Still no closer to displaying the symmetry as an integral but I just wanted to show some interesting consequences of the symmetry and the relation itself. Also, if we wanted to preserve the symmetry and generalise equation $(1)$ then we could do so by applying any symmetric differential operator as a raising operator e.g. ${\partial_{\alpha}}{\partial_{\beta}}$.

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Good work, Graham. I'll be interested very much to hear what you discover. –  Ron Gordon Nov 1 '13 at 18:45

Not an answer, but a reply to Maesumi's comment --

Invariance under $\alpha \to -\alpha$ is not so difficult to see: let $x=y^{-1}$, then \begin{align*}\int_0^{\infty} \frac{x^{\alpha}}{1+2x\cos(\pi \beta) + x^2} dx &= \int_{\infty}^0 \frac{y^{-\alpha} }{1+2y^{-1}\cos(\pi\beta)+y^{-2}}(-y^{-2})dy\\ &=\int_0^{\infty} \frac{y^{-\alpha}}{1+2y\cos(\pi \beta) + y^2} dy. \end{align*}

No idea about $\alpha \leftrightarrow \beta$ though.

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This seems like progress, but it is still far from clear what it buys us. By using this $\alpha \leftrightarrow - \alpha$ symmetry, we can get a cosine-like structure in the numerator (by adding contributions from the equal $\alpha$/$-\alpha$ pieces). But I am still stuck. –  Ron Gordon Jan 2 '13 at 16:51

$\newcommand{\+}{^{\dagger}}% \newcommand{\angles}[1]{\left\langle #1 \right\rangle}% \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}% \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}% \newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}% \newcommand{\dd}{{\rm d}}% \newcommand{\down}{\downarrow}% \newcommand{\ds}[1]{\displaystyle{#1}}% \newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}}% \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}% \newcommand{\fermi}{\,{\rm f}}% \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}% \newcommand{\half}{{1 \over 2}}% \newcommand{\ic}{{\rm i}}% \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow}% \newcommand{\isdiv}{\,\left.\right\vert\,}% \newcommand{\ket}[1]{\left\vert #1\right\rangle}% \newcommand{\ol}[1]{\overline{#1}}% \newcommand{\pars}[1]{\left( #1 \right)}% \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}}% \newcommand{\root}[2][]{\,\sqrt[#1]{\,#2\,}\,}% \newcommand{\sech}{\,{\rm sech}}% \newcommand{\sgn}{\,{\rm sgn}}% \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}}% \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ $\ds{\fermi\pars{\alpha,\beta} = \int_{0}^{\infty}\dd x\, {x^{\alpha} \over 1 + 2x\cos\pars{\pi\beta} + x^{2}} = \fermi\pars{\beta,\alpha}:\ {\Large ?}}$.$\quad\alpha, \beta \in \pars{-1,1}$.

Let's consider the integral $\ds{\fermi\pars{\alpha,\beta,\Lambda}\equiv\int_{0}^{\Lambda} {x^{\alpha} \over 1 + 2\cos\pars{\pi\beta}x + x^{2}}\,\dd x}$, with $\Lambda > 0$, such that $\ds{\fermi\pars{\alpha,\beta} = \lim_{\Lambda \to \infty}\fermi\pars{\alpha,\beta\Lambda}}$.

Note that \begin{align} &1 + 2x\cos\pars{\pi\beta} + x^{2} =\bracks{x + \cos\pars{\pi\beta}}^{2} + \sin^{2}\pars{\beta} \\[3mm]&=\bracks{x + \cos\pars{\pi\beta} + \ic\sin\pars{\pi\beta}} \bracks{x + \cos\pars{\pi\beta} - \ic\sin\pars{\pi\beta}} =\pars{x + \expo{\ic\pi\verts{\beta}}}\pars{x + \expo{-\ic\pi\verts{\beta}}} \end{align}

\begin{align} \fermi\pars{\alpha,\beta,\Lambda}&=\int_{0}^{\Lambda}x^{\alpha} \pars{{1 \over x + \expo{-\ic\pi\verts{\beta}}} - {1 \over x + \expo{\ic\pi\verts{\beta}}}} \,{1 \over \expo{\ic\pi\verts{\beta}} - \expo{-\ic\pi\verts{\beta}}}\,\dd x \\[3mm]&={1 \over 2\ic\sin\pars{\pi\verts{\beta}}}\bracks{% \int_{0}^{\Lambda}{x^{\alpha} \over x + \expo{-\ic\pi\verts{\beta}}}\,\dd x - \int_{0}^{\Lambda}{x^{\alpha} \over x + \expo{\ic\pi\verts{\beta}}}\,\dd x} \\[3mm]&= {1 \over 2\ic\sin\pars{\pi\verts{\beta}}}\bracks{% \expo{-\ic\pi\alpha\verts{\beta}} \int_{0}^{\Lambda\expo{\ic\pi\verts{\beta}}}{x^{\alpha} \over x + 1}\,\dd x - \expo{\ic\pi\alpha\verts{\beta}} \int_{0}^{\Lambda\expo{-\ic\pi\alpha\verts{\beta}}}{x^{\alpha} \over x + 1}\,\dd x} \\[3mm]&={1 \over 2\ic\sin\pars{\pi\beta}}\times \\[3mm]&\left\lbrace% \expo{-\ic\pi\alpha\verts{\beta}}\bracks{% \int_{0}^{\Lambda}{x^{\alpha} \over x + 1}\,\dd x + \int_{0}^{\Lambda\sin\pars{\pi\verts{\beta}}} {\bracks{\Lambda\cos\pars{\pi\beta} + \ic y}^{\alpha} \over \Lambda\cos\pars{\pi\beta} + \ic y + 1}\,\ic\,\dd y}\right. \\[3mm]&\phantom{\braces{}}- \\[3mm]&\phantom{\braces{}}\left.% \expo{\ic\pi\alpha\verts{\beta}}\bracks{% \int_{0}^{\Lambda}{x^{\alpha} \over x + 1}\,\dd x + \int_{0}^{-\Lambda\sin\pars{\pi\verts{\beta}}} {\bracks{\Lambda\cos\pars{\pi\beta} + \ic y}^{\alpha} \over \Lambda\cos\pars{\pi\beta} + \ic y + 1}\,\ic\,\dd y}\,\right\rbrace \end{align}

\begin{align} \fermi\pars{\alpha,\beta,\Lambda}&= -\,{\sin\pars{\pi\alpha\beta} \over \sin\pars{\pi\beta}} \int_{0}^{\Lambda}{x^{\alpha} \over x + 1}\,\dd x \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\,\,\pars{1} \\[3mm]&\phantom{=}+ {1 \over \sin\pars{\pi\verts{\beta}}}\Re\bracks{\expo{-\ic\pi\alpha\verts{\beta}}% \int_{0}^{\Lambda\sin\pars{\pi\verts{\beta}}} {\bracks{\Lambda\cos\pars{\pi\beta} + \ic y}^{\alpha} \over \Lambda\cos\pars{\pi\beta} + \ic y + 1}\,\dd y}\qquad\qquad\pars{2} \end{align}

$$ \mbox{When}\ \Lambda \to \infty\,,\quad\left\lbrace% \begin{array}{rl} \bullet & \mbox{The integral in}\ \pars{1}\ \mbox{converges when}\ \alpha < 0.\ \mbox{Since}\ \alpha\in\pars{-1,1}\,, \\&\mbox{the result is valid when}\ \alpha\in\pars{-1,0} \\[3mm] \bullet & \mbox{The integral in}\ \pars{2}\ \mbox{vanishes out in this limit.} \end{array}\right. $$ Then, $$ \lim_{\Lambda \to \infty}\fermi\pars{\alpha,\beta,\Lambda} =-\,{\sin\pars{\pi\alpha\beta} \over \sin\pars{\pi\beta}}\int_{0}^{\infty} {x^{\alpha} \over x + 1}\,\dd x\,,\qquad -1 < a < 0 $$

$$ \mbox{Also,}\quad {1 \over \sin\pars{\pi\beta}}= -\,{1 \over \pi}\int_{0}^{\infty}{y^{\beta} \over y + 1}\,\dd y\,,\qquad -1 < \beta < 0\tag{3} $$ $\tt @user7530$ has already proved the original integral symmetry under $\alpha \to -\alpha$ and it is clearly symmetric under $\beta \to -\beta$. The cases $\alpha = 0$ or $\beta = 0$ are easily handled $\pars{~\mbox{see}\ \pars{3}~}$ with the limit $$ \lim_{\mu \to 0^{-}}\bracks{\mu\int_{0}^{\infty}{x^{\mu} \over x + 1}}\,\dd x = -1\tag{4} $$

$$ \fermi\pars{\alpha,\beta} = {\sin\pars{\pi\verts{\alpha\beta}} \over \pi}\int_{0}^{\infty}\int_{0}^{\infty} {x^{-\verts{\alpha}}y^{-\verts{\beta}} \over xy + x + y + 1}\,\dd x\,\dd y \quad\mbox{if}\quad\alpha, \beta \in \pars{-1,0}\cup\pars{0,1} $$ Otherwise, $$ \fermi\pars{\mu,0} = \fermi\pars{0,\mu} =\lim_{\nu \to 0^{-}}\fermi\pars{\mu,\nu} =\lim_{\mu \to 0^{-}}\fermi\pars{\mu,\nu} $$

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However I find some issues with signs. if $\beta \in (0, 1)$ and $y \in (0, \infty)$ then $\dfrac{y^{\beta}}{y + 1} > 0$ so that $\int_{0}^{\infty}(y^{\beta})/(y + 1)\,dy > 0$, but you have put that as $-\pi/\sin(\pi\beta)$. I believe you should get $f(\alpha, \beta) = \dfrac{\sin(\pi\alpha\beta)}{\sin(\pi\beta)}\int_{0}^{\infty}\dfrac{x^{\alpha}}{‌​x + 1}\,dx$ because here also we don't need a $-$ sign if we restrict $\alpha \in (0, 1), \beta \in (0, 1)$. –  Paramanand Singh Feb 16 at 10:55
The right formula is $\int_{0}^{\infty}(y^{-\beta})/(y + 1)\, dy = \pi/\sin(\pi\beta)$ where $0 < \beta < 1$. Perhaps there is some issue in the last few lines. –  Paramanand Singh Feb 16 at 11:30
@ParamanandSingh I'll check and comment later. Some little issues will be clear. Right now this page is working very bad. Thanks. –  Felix Marin Feb 16 at 22:25
@ParamanandSingh I just checked every thing and rewrote the answer to deal with the worries you commented about it. Thanks. –  Felix Marin Feb 17 at 16:47

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