Computing the conical function Pμ-1/2+ir(x)
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A stable computational scheme for the conical function Pμ− 1/2+iτ (x) for x > −1, real τ, and μ ≤ 0 or μ ∈ N is presented. The scheme combines uniform asymptotic expansions for large |μ| with the application of the three-term recurrence relation on the μ index in the direction of decreasing |μ| when x > 0. When x < 0, the conditioning of recursion is the opposite, and conical functions can be computed in the direction of increasing |μ|.