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
χ ν ( z ) = ∑ k = 0 ∞ z 2 k + 1 ( 2 k + 1 ) ν . {\displaystyle \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 as the odd part of the polylogarithm
χ ν ( z ) = 1 2 [ Li ν ( z ) − Li ν ( − z ) ] . {\displaystyle \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
χ ν ( z ) = 2 − ν z Φ ( z 2 , ν , 1 / 2 ) . {\displaystyle \chi _{\nu }(z)=2^{-\nu }z\,\Phi (z^{2},\nu ,1/2).}
Identities
χ − 1 ( x ) = x ( 1 + x 2 ) ( 1 − x 2 ) 2 {\displaystyle \chi _{-1}\left(x\right)={\frac {x\left(1+x^{2}\right)}{\left(1-x^{2}\right)^{2}}}}
χ 0 ( x ) = x 1 − x 2 {\displaystyle \chi _{0}\left(x\right)={\frac {x}{1-x^{2}}}}
χ 1 ( x ) = arctanh ( x ) = 1 2 ln ( 1 + x 1 − x ) {\displaystyle \chi _{1}(x)=\operatorname {arctanh} (x)={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)}
χ 2 ( x ) = ∫ 0 x arctanh ( t ) t d t {\displaystyle \chi _{2}(x)=\int _{0}^{x}{\frac {\operatorname {arctanh} (t)}{t}}\mathrm {d} t}
χ ∞ ( x ) = x {\displaystyle \chi _{\infty }(x)=x}
d d x χ ν ( x ) = χ ν − 1 ( x ) x {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\chi _{\nu }(x)={\frac {\chi _{\nu -1}(x)}{x}}}
d d x χ 2 ( x ) = arctanh ( x ) x {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\chi _{2}(x)={\frac {\operatorname {arctanh} (x)}{x}}}
χ 2 ( x ) + χ 2 ( 1 / x ) = { 1 4 ( ln 2 ( x ) − ln 2 ( − x ) ) for | x | > 1 1 4 ( ln 2 ( 1 / x ) − ln 2 ( − 1 / x ) ) for | x | ≤ 1 ∖ { 0 } {\displaystyle \chi _{2}(x)+\chi _{2}(1/x)=\left\{{\begin{array}{rcl}{\frac {1}{4}}{\bigl (}\ln ^{2}(x)-\ln ^{2}(-x){\bigr )}&{\text{for}}&|x|>1\\[4pt]{\frac {1}{4}}{\bigl (}\ln ^{2}(1/x)-\ln ^{2}(-1/x){\bigr )}&{\text{for}}&|x|\leq 1\setminus \{0\}\end{array}}\right.}
χ 2 ( x ) + χ 2 ( 1 − x 1 + x ) = π 2 8 + ln ( x ) arctanh ( x ) , x ∈ ( 0 , 1 ) {\displaystyle \chi _{2}(x)+\chi _{2}\left({\frac {1-x}{1+x}}\right)={\frac {\pi ^{2}}{8}}+\ln(x)\operatorname {arctanh} (x),\quad x\in (0,1)}
Special Values It takes the special values:
χ 2 ( 1 ) = π 2 8 {\displaystyle \chi _{2}(1)={\frac {\pi ^{2}}{8}}}
χ 2 ( − 1 ) = − π 2 8 {\displaystyle \chi _{2}(-1)=-{\frac {\pi ^{2}}{8}}}
χ 2 ( 2 − 1 ) = π 2 16 − 1 4 ln 2 ( 2 + 1 ) {\displaystyle \chi _{2}({\sqrt {2}}-1)={\frac {\pi ^{2}}{16}}-{\frac {1}{4}}\ln ^{2}({\sqrt {2}}+1)}
χ 2 ( 5 − 1 2 ) = π 2 12 − 3 4 ln 2 ( 5 + 1 2 ) {\displaystyle \chi _{2}\left({\frac {{\sqrt {5}}-1}{2}}\right)={\frac {\pi ^{2}}{12}}-{\frac {3}{4}}\ln ^{2}\left({\frac {{\sqrt {5}}+1}{2}}\right)}
χ 2 ( 5 − 2 ) = π 2 24 − 3 4 ln 2 ( 5 + 1 2 ) {\displaystyle \chi _{2}({\sqrt {5}}-2)={\frac {\pi ^{2}}{24}}-{\frac {3}{4}}\ln ^{2}\left({\frac {{\sqrt {5}}+1}{2}}\right)}
χ 2 ( i ) = i K , {\displaystyle \chi _{2}(i)=iK,}
where i i is the imaginary unit and K is Catalan's constant. Other special values include:
χ n ( 1 ) = λ ( n ) {\displaystyle \chi _{n}(1)=\lambda (n)}
χ n ( i ) = i β ( n ) , {\displaystyle \chi _{n}(i)=i\beta (n),}
where λ ( n ) \lambda (n) is the Dirichlet lambda function and β ( n ) \beta (n) is the Dirichlet beta function.
Integral relations
∫ 0 π / 2 arcsin ( r sin θ ) d θ = χ 2 ( r ) , ∫ 0 π / 2 arccos ( r cos θ ) d θ = ( π 2 ) 2 − χ 2 ( r ) i f | r | ≤ 1 {\displaystyle \int _{0}^{\pi /2}\arcsin(r\sin \theta )\,\mathrm {d} \theta =\chi _{2}(r),\qquad \int _{0}^{\pi /2}\arccos(r\cos \theta )\,\mathrm {d} \theta =\left({\frac {\pi }{2}}\right)^{2}-\chi _{2}(r)\qquad {\rm {if}}~~|r|\leq 1}
∫ 0 π / 2 arctan ( r sin θ ) d θ = − 1 2 ∫ 0 π r θ cos θ 1 + r 2 sin 2 θ d θ = 2 χ 2 ( 1 + r 2 − 1 r ) , r ≠ 0 {\displaystyle \int _{0}^{\pi /2}\arctan(r\sin \theta )\,\mathrm {d} \theta =-{\frac {1}{2}}\int _{0}^{\pi }{\frac {r\theta \cos \theta }{1+r^{2}\sin ^{2}\theta }}\,\mathrm {d} \theta =2\chi _{2}\!\!\left({\frac {{\sqrt {1+r^{2}}}-1}{r}}\right),\quad r\neq 0}
∫ 0 π / 2 arctan ( p sin θ ) arctan ( q sin θ ) d θ = π χ 2 ( 1 + p 2 − 1 p ⋅ 1 + q 2 − 1 q ) , p , q ≠ 0 {\displaystyle \int _{0}^{\pi /2}\arctan(p\sin \theta )\arctan(q\sin \theta )\,\mathrm {d} \theta =\pi \chi _{2}\!\!\left({\frac {{\sqrt {1+p^{2}}}-1}{p}}\cdot {\frac {{\sqrt {1+q^{2}}}-1}{q}}\right),\quad p,q\neq 0}
∫ 0 π / 2 arctanh ( p sin θ ) arctanh ( q sin θ ) d θ = π χ 2 ( 1 − 1 − p 2 p ⋅ 1 − 1 − q 2 q ) , | p | , | q | ≤ 1 ∖ { 0 } {\displaystyle \int _{0}^{\pi /2}\operatorname {arctanh} (p\sin \theta )\operatorname {arctanh} (q\sin \theta )\,\mathrm {d} \theta =\pi \chi _{2}\!\!\left({\frac {1-{\sqrt {1-p^{2}}}}{p}}\cdot {\frac {1-{\sqrt {1-q^{2}}}}{q}}\right),\quad |p|,|q|\leq 1\setminus \{0\}}
∫ 0 α ∫ 0 β d x d y 1 − x 2 y 2 = χ 2 ( α β ) i f | α β | ≤ 1 {\displaystyle \int _{0}^{\alpha }\!\!\int _{0}^{\beta }{\frac {\mathrm {d} x\,\mathrm {d} y}{1-x^{2}y^{2}}}=\chi _{2}(\alpha \beta )\qquad {\rm {if}}~~|\alpha \beta |\leq 1}
References
Weisstein, Eric W. "Legendre's Chi Function". MathWorld. Djurdje Cvijović, Jacek Klinowski (1999). "Values of the Legendre chi and Hurwitz zeta functions at rational arguments". Mathematics of Computation. 68 (228): 1623–1630. doi:10.1090/S0025-5718-99-01091-1. Djurdje Cvijović (2007). "Integral representations of the Legendre chi function". Journal of Mathematical Analysis and Applications. 332 (2): 1056–1062. arXiv:0911.4731. doi:10.1016/j.jmaa.2006.10.083. S2CID 115155704.