Using the Cauchy integral theorem, we develop the application of the steepest descent method to efficiently compute the three-dimensional acoustic single-layer integral operator for large wave numbers. Explicit formulas for the splitting points are derived in the one-dimensional case to build suitable complex integration paths. The construction of admissible steepest descent paths is next investigated and some of their properties are stated. Based on these theoretical results, we derive the quadrature scheme of the oscillatory integrals first in dimension one and then extend the methodology to three-dimensional planar triangles. Numerical simulations are finally reported to illustrate the accuracy and efficiency of the proposed approach.

中文翻译：

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高频区高效计算三维声学单层算子的最速下降法分析

利用柯西积分定理，我们开发了最速下降法的应用，以有效地计算大波数的三维声学单层积分算子。在一维情况下导出分裂点的显式公式以构建合适的复杂积分路径。接下来研究允许的最陡下降路径的构造，并说明它们的一些性质。基于这些理论结果，我们首先在第一维推导了振荡积分的正交方案，然后将该方法扩展到三维平面三角形。最后报告了数值模拟，以说明所提出方法的准确性和效率。