Functions which can be expressed in terms of Legendre functions of the first and second kinds. See Abramowitz and Stegun. P_(-1/2 + i p)(cos θ) | = | 1 + (4p^2 + 1^2)/2^2 sin^2(1/2 θ) + ((4p^2 + 1^2)(4p^2 + 3^2))/(2^2 4^2) sin^4(1/2 θ) + ... | = | 2/π integral_0^θ (cosh(p t) d t)/sqrt(2(cos t - cos θ)) Q_(-1/2 ∓ i p)(cos θ) | = | ± i sinh(p π) integral_0^∞ (cos(p t) d t)/sqrt(2(cosh t + cos θ)) + integral_0^∞ (cosh(p t) d t)/sqrt(2(cos t - cos θ)).
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