Legendre chi function
Template:Short description 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 polynomials, 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>