用初等微分方程和围道积分进行傅里叶变换反演
Fourier transform inversion using an elementary differential equation and a contour integral
摘要：
设$f$为实数直线上的函数。在$f$是绝对连续的，使得$f$和$f'$是Lebesgue可积的假设下，证明了傅里叶变换反演定理.函数$g$由$f'(t)-iwf(t)=g(t)$定义。该微分方程具有公知的使用Heaviside阶梯函数的积分解。用留数的初等计算将Heaviside阶梯函数写成简单的围道积分。其余的证明需要对积分进行初等处理。因此，傅里叶变换反演定理证明与很少的机器。只有微小的变化的方法也被用来证明反演定理的职能的几个变量和证明黎曼的本地化定理。
英文摘要：
Let $f$ be a function on the real line. The Fourier transform inversion theorem is proved under the assumption that $f$ is absolutely continuous such that $f$ and $f'$ are Lebesgue integrable. A function $g$ is defined by $f'(t)-iwf(t)=g(t)$. This differential equation has a well known integral solution using the Heaviside step function. An elementary calculation with residues is used to write the Heaviside step function as a simple contour integral. The rest of the proof requires elementary manipulation of integrals. Hence, the Fourier transform inversion theorem is proved with very little machinery. With only minor changes the method is also used to prove the inversion theorem for functions of several variables and to prove Riemann's localization theorem.
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