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18 Orthogonal PolynomialsApplications18.39 Applications in the Physical Sciences
Figure 18.39.2 (See in context.)
See accompanying text
Figure 18.39.2: Coulomb–Pollaczek weight functions, x∈[−1,1], (18.39.50) for s=10, l=0, and Z=±1. For Z=+1 the weight function, red curve, has an essential singularity at x=−1, as all derivatives vanish as x→−1+; the green curve is ∫−1xwCP⁡(y)⁢dy, to be compared with its histogram approximation in §18.40(ii). For Z=−1 the weight function, blue curve, is non-zero at x=−1, but this point is also an essential singularity as the discrete parts of the weight function of (18.39.51) accumulate as k→∞, xk→−1−.
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