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Legendre chi function
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{{Short description|Mathematical Function}} In [[mathematics]], the '''Legendre chi function''' is a [[special function]] whose [[Taylor series]] is also a [[Dirichlet series]], given by <math display="block">\chi_\nu(z) = \sum_{k=0}^\infty \frac{z^{2k+1}}{(2k+1)^\nu}.</math> As such, it resembles the Dirichlet series for the [[polylogarithm]], and, indeed, is trivially expressible in terms of the polylogarithm as <math display="block">\chi_\nu(z) = \frac{1}{2}\left[\operatorname{Li}_\nu(z) - \operatorname{Li}_\nu(-z)\right].</math> The Legendre chi function appears as the [[discrete Fourier transform]], with respect to the order ν, of the [[Hurwitz zeta function]], and also of the [[Euler polynomial]]s, with the explicit relationships given in those articles. The Legendre chi function is a special case of the [[Lerch transcendent]], and is given by <math display="block">\chi_\nu(z)=2^{-\nu}z\,\Phi (z^2,\nu,1/2).</math> ==Identities== <math display="block">\chi_2(x) + \chi_2(1/x)= \frac{\pi^2}{4}-\frac{i \pi}{2}\ln |x| .</math> <math display="block">\frac{d}{dx}\chi_2(x) = \frac{{\rm arctanh\,} x}{x}.</math> ==Integral relations== <math display="block">\int_0^{\pi/2} \arcsin (r \sin \theta) d\theta = \chi_2\left(r\right)</math> <math display="block">\int_0^{\pi/2} \arctan (r \sin \theta) d\theta = -\frac{1}{2}\int_0^{\pi} \frac{ r \theta \cos \theta}{1+ r^2 \sin^2 \theta} d\theta = 2 \chi_2\left(\frac{\sqrt{1+r^2}- 1}{r}\right)</math> <math display="block">\int_0^{\pi/2} \arctan (p \sin \theta)\arctan (q \sin \theta) d\theta = \pi \chi_2\left(\frac{\sqrt{1+p^2}- 1}{p}\cdot\frac{\sqrt{1+q^2}- 1}{q}\right)</math> <math display="block">\int_0^{\alpha}\int_0^{\beta} \frac{dx dy}{1-x^2 y^2} = \chi_2(\alpha\beta)\qquad {\rm if}~~|\alpha\beta|\leq 1</math> ==References== * {{mathworld|urlname=LegendresChi-Function |title=Legendre's Chi Function}} * {{cite journal|author= Djurdje Cvijović, Jacek Klinowski|year= 1999|title=Values of the Legendre chi and Hurwitz zeta functions at rational arguments|journal= Mathematics of Computation|volume= 68|issue= 228|pages= 1623–1630|doi=10.1090/S0025-5718-99-01091-1|doi-access=free }} * {{note_label|Cvijovic2006||}}{{cite journal|author=Djurdje Cvijović|year= 2007 |title=Integral representations of the Legendre chi function|journal= Journal of Mathematical Analysis and Applications |volume= 332|issue= 2|pages= 1056–1062|doi=10.1016/j.jmaa.2006.10.083|arxiv=0911.4731|s2cid= 115155704 }} [[Category:Special functions]] {{mathanalysis-stub}}
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