Mathematics of Computation
In a previous discussion of a paper by Carrus and Treuenfels (CT), a difference test indicated that some of the early zeros of the associated Legendre function Pn^cos 0) = 0 as a function of n were incorrect (MTAC, v. 5,. The investigation of the present article also reveals some errors. The authors give an alternative proof of an equation due to Macdonald2 for determining the early zeros of P"m(cos 0) = 0 where 0 is near t. For m = 1, 6 = 165°, this formula gives 1.035 as an approximation to
... approximation to the first zero. Employing power series, it is shown that the first zero must be between 1.0316 and 1.0321. The value reported by CT is 1.053 and so is in error. Application of the Macdonald formula shows that for 165° < 6 < 180°, the corresponding values of « decrease with increasing 0. For m = 1, 0 = 170°, the first zero given by CT is 1.05 and thus is also incorrect. Numerical analysis of early zeros for values of 6 other than those cited above is not given, but sufficient evidence now exists to show that the CT tables should be used with caution.