τ-methodによる複素変数のベッセル関数K_n(z)の数値計算

概要

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In this paper we describe a method for the calculation of K_n(z) with complex argument. The τ-method proposed by C.Lanczos is applied to the computation of K_n(z). The function K_n(z) is written K_n(z)=(π/(2π))^<1/2>e^<-n>f_n(1/z), where f_n(t) satisfies the differential equation t^2f_n"(t)+2(t+1)f_n'(t)-(n^2-(1/4))f_n(t)=0. The polynomial approximation for f_n(t) is performed by putting the shifted Ultraspherical polynomial to the right hand side of this equation. Then it is found from error analysis that the choice of the shifted Legendre polynomial-a special case of the Ultraspherical polynomial-leads to the most efficient computation of f_n(t) for Re(z)>0. Combining this method with the method using the recurrence technique, the value of K_n(z) can be obtained efficiently in the whole complex region.
一般社団法人情報処理学会の論文
1973-08-15