留数定理与傅里叶变换的实验例证式教学研究——以一维散射问题为例

Case-Based Experimental Teaching of the Residue Theorem and Fourier Transform—A One-Dimensional Scattering Example

摘要: 在《复变函数与积分变换》课程中，学生通常能够完成留数计算和傅里叶变换的公式运算，却难以把这两部分知识联系起来，也不清楚它们在实际问题中如何配合使用。本课程为面向工科各专业开设的公共基础课，本方案适用于大学二年级上学期修读该课程的本科生。学生应已学习复数、留数定理和傅里叶变换的基本性质，但不要求预先掌握格林函数和散射理论。针对这一现象，本文选取一维弱散射问题，设计相应的实验教学案例。首先，将波动方程积分表示中的核函数写成傅里叶逆变换，并通过围道积分和留数定理求得其解析式；其次，在弱散射的一阶近似下，建立反射数据与目标函数傅里叶频谱的对应关系，并利用有限频带数据近似恢复目标；最后，通过改变目标的位置、宽度和间距，以及测量带宽、噪声和相位信息，观察计算结果的变化并分析原因。教学中不展开一维散射和格林函数的一般理论，而把简化模型作为留数定理、傅里叶变换性质和逆变换的综合应用。本文给出相应的教学内容、实验任务和评价方案，为上述知识点的衔接教学提供一个可实施的案例。

Abstract: In teaching the course Complex Functions and Integral Transforms, students are often able to compute residues and carry out routine Fourier-transform calculations, yet they have difficulty understanding how the two topics are connected and how they can be used together in practical problems. The course is a common foundational course for students from a range of engineering majors. The proposed unit is intended for undergraduates taking the course in the first semester of their sophomore year. Students are expected to have studied complex numbers, the residue theorem, and basic properties of the Fourier transform; no prior knowledge of Green’s functions or scattering theory is required. To address this difficulty, this paper develops a teaching case around a simplified one-dimensional weak-scattering problem. First, the integral kernel associated with the wave equation is expressed as an inverse Fourier transform, and its closed-form expression is derived using contour integration and the residue theorem. Next, under a first-order approximation for weak scattering, the relationship between the reflection coefficient and the Fourier spectrum of the target profile is established, and the target is approximately reconstructed from finite-bandwidth reflection data. Numerical experiments are conducted by varying the target position, width, and separation, as well as the measurement bandwidth and noise level, and by comparing the results obtained with and without phase information. These experiments allow students to observe changes in the computed results and explain their causes. Rather than introducing the general theories of one-dimensional scattering and Green’s functions, the teaching design uses the simplified model to bring together the residue theorem, properties of the Fourier transform, and the inverse Fourier transform in a single application. The paper presents the teaching content, experimental tasks, and assessment plan, providing a practical example for connecting these topics in classroom instruction.