imported>GünniX: Reflist

2025-12-19T01:25:40Z

Reflist

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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
+In [[mathematics]], the '''Legendre chi function''' (named after [[Adrien-Marie Legendre]]) 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
@@ Line 14: / Line 15: @@
 <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==
@@ Line 26: / Line 52: @@
 ==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 }}
@@ Line 34: / Line 61: @@
 [[Category:Special functions]]
 {{mathanalysis-stub}}

87.157.178.34: /* Integral relations */ Added the complementary integral of the cosine to the first relation for completeness

2025-06-15T10:44:50Z

Integral relations: Added the complementary integral of the cosine to the first relation for completeness

New page

{{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),\qquad\int_0^{\pi/2} \arccos (r \cos \theta) d\theta
= \left(\frac{\pi}{2}\right)^2-\chi_2\left(r\right)\qquad {\rm if}~~|r|\leq 1</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}}

Legendre chi function - Revision history

imported>GünniX: Reflist

87.157.178.34: /* Integral relations */ Added the complementary integral of the cosine to the first relation for completeness