Real gas

In mathematics, the Legendre chi function is a special function whose Taylor series is also a Dirichlet series, given by

$\chi _{\nu }(z)=\sum _{k=0}^{\infty }{\frac {z^{2k+1}}{(2k+1)^{\nu }}}.$

As such, it resembles the Dirichlet series for the polylogarithm, and, indeed, is trivially expressible in terms of the polylogarithm as

$\chi _{\nu }(z)={\frac {1}{2}}\left[\operatorname {Li} _{\nu }(z)-\operatorname {Li} _{\nu }(-z)\right].$

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

$\chi _{n}(z)=2^{-n}z\,\Phi (z^{2},n,1/2).\,$

Identities

$\chi _{2}(x)+\chi _{2}(1/x)={\frac {\pi ^{2}}{4}}-{\frac {i\pi }{2}}|\ln x|\qquad (x>0).$

${\frac {d}{dx}}\chi _{2}(x)={\frac {{\rm {arctanh\,}}x}{x}}.$

Integral relations

$\int _{0}^{\pi /2}\arctan(r\sin \theta )d\theta =-{\frac {1}{2}}\int _{0}^{\pi }{\frac {r\theta \cos \theta }{1+r^{2}\sin ^{2}\theta }}d\theta =2\chi _{2}\left({\frac {{\sqrt {1+r^{2}}}-1}{r}}\right)$

$\int _{0}^{\pi /2}\arctan(p\sin \theta )\arctan(q\sin \theta )d\theta =\pi \chi _{2}\left({\frac {{\sqrt {1+p^{2}}}-1}{p}}\cdot {\frac {{\sqrt {1+q^{2}}}-1}{q}}\right)$

$\int _{0}^{\alpha }\int _{0}^{\beta }{\frac {dxdy}{1-x^{2}y^{2}}}=\chi _{2}(\alpha \beta )\qquad {\rm {if}}~~|\alpha \beta |\leq 1$

References

• 22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.
  
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• Djurdje Cvijović and Jacek Klinowski, "Values of the Legendre chi and Hurwitz zeta functions at rational arguments", Mathematics of Computation 68 (1999), 1623-1630.
• 24 years old Art Manager or Manager Harry Kamm from West Hill, really likes ceramics, property developers in Singapore house for rent and walking. Recently has traveled to .Template:Cite web
• Mathematics Stack Exchange