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~P_{-\frac{1}{2}}(z)

~=

~
\frac{2}{\pi} \biggl[\frac{2}{z+1}\biggr]^{1 / 2} ~K\biggl( \sqrt{ \frac{z-1}{z+1}} \biggr)

      for example …

~P_{-\frac{1}{2}}(\cosh\eta)

~=

~
\biggl[ \frac{\pi}{2} \cdot \cosh \frac{\eta}{2} \biggr]^{-1} K\biggl( \tanh \frac{\eta}{2} \biggr)

Abramowitz & Stegun (1995), p. 337, eq. (8.13.1)

Abramowitz & Stegun (1995), p. 337, eq. (8.13.2)

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