关于Jacobi函数的渐近性态研究

采取改进取点x(t)的做法,提高了Jacobi函数的一项近似精确度.我们分别取x(t)的两项和三项,做出了Jacobi函数φ(α,β)μ(t) (α>-1 )当μ→+∞渐近近似,并给出了相应的误差限.随着x(t)取的项数增加,即点x(t)取的更“精确”,Jacobi函数φ(α,β)μ(t)渐近近似的精确度也随之提高.

Study of Asymptotic Properties for Jacobi Functions

The one-term approximation of Jacobi functions φ (α,β)_μ(t) can be made more accurate by choosing the optimum x(t) location instead of increasing the number of terms in the asymptotic approximation. Respectively we take one term or two terms of x(t), obtain an asymptotic expansion Jacobi functions φ (α,β)_μ(t)α>-1, as μ→+∞, and give corresponding error bounds. The error in the one-term approximation of φ (α,β)_μ(t) can be made arbitrarily small by using enough terms for x(t).