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

~=

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

      for example …

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

~=

~
\frac{2}{\pi}~e^{\eta/2}~ E( \sqrt{1-e^{-2\eta}} )

Abramowitz & Stegun (1995), p. 337, eq. (8.13.5)

Abramowitz & Stegun (1995), p. 337, eq. (8.13.6)

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