3

$$\int_{0}^{1} \frac{x^a+x^{-a}}{1+x^2} dx$$

where $a \in (-1,1)$

This is part from a bunch of residue theorem exercises and I can't find out how this can relate to complex analysis. Some common substitutions might solve this in the real way but how to approach this in the other way?

SADBOYS
  • 1,219

3 Answers3

5

Use$$\int_0^1\frac{x^{-a}dx}{1+x^2}=\int_1^\infty\frac{y^ady}{1+y^2}$$to rewrite the integral as$$\int_0^\infty\frac{x^adx}{1+x^2}.$$Now use $x=\tan t$ to rewrite it again as$$\int_0^{\pi/2}\tan^atdt=\frac12\operatorname{B}\left(\frac{1+a}{2},\,\frac{1-a}{2}\right)=\frac{\pi}{2}\sec\frac{a\pi}{2}.$$While most of this uses Beta & Gamma functions, you also need the Gamma function's reflection formula, which has proofs using complex analysis. Alternatively, we could replace $x=\tan t$ with $x=\exp u$, then continue thereafter with the residue theorem.

J.G.
  • 115,835
5

Cosecant Series $$ \begin{align} \int_0^1\frac{x^a+x^{-a}}{1+x^2}\,\mathrm{d}x &=\scriptsize\left(\frac1{a+1}-\frac1{a-1}\right)-\left(\frac1{a+3}-\frac1{a-3}\right)+\left(\frac1{a+5}-\frac1{a-5}\right)\cdots\tag1\\ &=\frac12\sum_{k\in\mathbb{Z}}\frac{-1^k}{k+\frac{a+1}2}\tag2\\ &=\frac\pi2\csc\left(\pi\frac{a+1}2\right)\tag3\\[6pt] &=\frac\pi2\sec\left(\frac{\pi a}2\right)\tag4 \end{align} $$ Explanation:
$(1)$: use series for $\frac1{1+x^2}$ and integrate term by term
$(2)$: write series as a sum over $\mathbb{Z}$; i.e. $\sum\limits_{k\in\mathbb{Z}}f(k)=\sum\limits_{k=1}^\infty(f(k-1)+f(-k))$
$(3)$: use this answer or this answer
$(4)$: $\csc\left(x+\frac\pi2\right)=\sec(x)$

Contour Integration $$ \begin{align} \int_0^1\frac{x^a+x^{-a}}{1+x^2}\,\mathrm{d}x &=\int_1^\infty\frac{x^a+x^{-a}}{1+x^2}\,\mathrm{d}x\tag5\\ &=\frac12\int_0^\infty\frac{x^a+x^{-a}}{1+x^2}\,\mathrm{d}x\tag6\\ &=\frac\pi4\left(\csc\left(\pi\frac{1+a}2\right)+\csc\left(\pi\frac{1-a}2\right)\right)\tag7\\[6pt] &=\frac\pi2\sec\left(\frac{\pi a}2\right)\tag8 \end{align} $$ Explanation:
$(5)$: substitute $x\mapsto\frac1x$
$(6)$: average the left and right hand sides of $(5)$
$(7)$: $(1)$ from this contour integration answer
$(8)$: $\csc\left(\frac\pi2+x\right)=\csc\left(\frac\pi2-x\right)=\sec(x)$

robjohn
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1

$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ It's a "candidate" to the Ramanujan's Master Theorem: \begin{align} &\bbox[5px,#ffd]{% \int_{0}^{1}{x^{a} + x^{-a} \over 1 + x^{2}}\,\dd x} = {1 \over 2}\int_{0}^{\infty}{x^{a} + x^{-a} \over 1 + x^{2}}\,\dd x \\[5mm] = &\ \int_{0}^{\infty}{x^{a} \over 1 + x^{2}}\,\dd x = {1 \over 2}\int_{0}^{\infty}{x^{\pars{\color{red}{a/2 + 1/2}} - 1} \over 1 + x}\,\dd x \end{align} Note that $\ds{{1 \over 1 + x} = \sum_{k = 0}^{\infty}\pars{-1}x^{k} = \sum_{k = 0}^{\infty}\color{red}{\Gamma\pars{k + 1}} \,{\pars{-x}^{k} \over k!}}$.

Then, \begin{align} &\bbox[5px,#ffd]{% \int_{0}^{1}{x^{a} + x^{-a} \over 1 + x^{2}}\,\dd x} = {1 \over 2}\,\Gamma\pars{{a \over 2} + {1 \over 2}} \Gamma\pars{-\bracks{{a \over 2} + {1 \over 2}} + 1} \\[5mm] = &\ {1 \over 2}\,{\pi \over\sin\pars{\pi a/2 + \pi/2}} = \bbx{{1 \over 2}\,\pi\sec\pars{\pi a \over 2}} \\ & \end{align}

Felix Marin
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