Let us teach this generalization of the final-value theorem
Let us teach this generalization of the final-value theorem
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让我们来教授终值定理的推广
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发表时间：
2003
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通讯作者：
E. Gluskin
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作者：
E. Gluskin
对工程数学基础课（或电路理论、自动控制等）中拉普拉斯变换的应用教学提出了建议。函数f(t)的终值定理只有在存在时才有意义。对于不存在但存在时间平均值的时间函数，将此定理推广为。这种推广包括周期函数或渐近周期函数，以及可以由周期函数的有限和给出的概周期函数。证明包括f(t)趋向于fas(t)指数的情况，这对于主要的物理和电路应用是现实的。将结果推广到离散序列，用z变换处理，简要地考虑。终值定理的广义形式应纳入工程数学课程。老师可以在课堂上引入有趣的新问题，并提供与（通常是稍后的）傅里叶级数和傅里叶变换的学习更好的联系。
A suggestion relevant to teaching the use of Laplace transforms in a basic course of engineering mathematics (or circuit theory, automatic control, etc) is made. The useful ‘final-value’ theorem for a function f(t), , , makes sense only if , , exists. A generalization of this theorem for time functions for which does not exist, but the time average exists, states that as , . This generalization includes the case of periodic or asymptotically periodic functions, and almost-periodic functions that can be given by finite sums of periodic functions. The proofs include the case of f(t) tending to fas(t) exponentially, which is realistic for the main physics and circuit applications. Extension of the results to discrete sequences, treatable by the z-transform, is briefly considered. The generalized form of the final-value theorem should be included in courses of engineering mathematics. The teacher can introduce interesting new problems into the lesson, and provide a better connection with the (usually later) study of the Fourier series and Fourier transform.