Difference between revisions of "Legendre chi"

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=Properties=
 
=Properties=
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[[Derivative of Legendre chi]]<br />
<strong>Proposition:</strong> The following formula holds for real $x>0$:
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[[Legendre chi in terms of polylogarithm]]<br />
$$\chi_2(x)+\chi_2 \left( \dfrac{1}{x} \right) = \dfrac{\pi^2}{4} - \dfrac{i\pi}{2}|\log x|.$$
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[[Catalan's constant using Legendre chi]]<br />
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<strong>Proof:</strong> █
 
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{{:Derivative of Legendre chi}}
 
 
 
{{:Legendre chi in terms of polylogarithm}}
 
{{:Catalan's constant using Legendre chi}}
 
  
 
=References=
 
=References=

Revision as of 07:59, 8 June 2016

The Legendre chi function $\chi_{\nu}$ is defined by $$\chi_{\nu}(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^{2k+1}}{(2k+1)^{\nu}}.$$

Properties

Derivative of Legendre chi
Legendre chi in terms of polylogarithm
Catalan's constant using Legendre chi

References

[1]