Legendre chi function: Difference between revisions

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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>

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==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 artanharctanh\,} 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/math>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

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==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]]