论文标题
某些广义正弦综合的渐进式
Asymptotics of some generalised sine-integrals
论文作者
论文摘要
我们通过利用saddle-saddle-points方法来获得通用正弦综合\ [\ int_0^\ infty \ left（\ frac {\ sin \，x} {x} {x} \ right）^{n} dx \ \]的渐近整数$ n $的渐近扩展。这种扩展显示与{\ it commun中的J. Schlage-puchta的最新结果一致。韩语数学。 Soc。} {\ bf 35}（2020）1193--1202谁使用了其他方法。 获得了另一个相关正弦综合的渐近估计，还涉及大量$ n $。给出数值结果以说明此近似值的准确性。我们还重新访问涉及Bessel函数$J_ν（x）$的Ball积分的渐近学，当$ν= 1/2 $时，它将减少到上述积分。
We obtain the asymptotic expansion for large integer $n$ of a generalised sine-integral \[\int_0^\infty\left(\frac{\sin\,x}{x}\right)^{n}dx\] by utilising the saddle-point method. This expansion is shown to agree with recent results of J. Schlage-Puchta in {\it Commun. Korean Math. Soc.} {\bf 35} (2020) 1193--1202 who used a different approach. An asymptotic estimate is obtained for another related sine-integral also involving a large power $n$. Numerical results are given to illustrate the accuracy of this approximation. We also revisit the asymptotics of Ball's integral involving the Bessel function $J_ν(x)$, which reduces to the above integral when $ν=1/2$.
