Difference between revisions of "Derivative of Legendre chi 2"

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==Theorem==
 
==Theorem==
 
The following formula holds:
 
The following formula holds:
$$\dfrac{\mathrm{d}}{\mathrm{d}x} \chi_2(x) = \dfrac{\mathrm{arctanh}(x)}{x},$$
+
$$\dfrac{\mathrm{d}}{\mathrm{d}z} \chi_2(z) = \dfrac{\mathrm{arctanh}(z)}{z},$$
 
where $\chi$ denotes the [[Legendre chi]] function and $\mathrm{arctanh}$ denotes the [[Arctanh|inverse hyperbolic tangent]] function.
 
where $\chi$ denotes the [[Legendre chi]] function and $\mathrm{arctanh}$ denotes the [[Arctanh|inverse hyperbolic tangent]] function.
  

Latest revision as of 01:31, 1 July 2017

Theorem

The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}z} \chi_2(z) = \dfrac{\mathrm{arctanh}(z)}{z},$$ where $\chi$ denotes the Legendre chi function and $\mathrm{arctanh}$ denotes the inverse hyperbolic tangent function.

Proof

References