考慮線性時滯微分系統，若由狀態轉移函數擬合時間響應，因是由一段段時間積分而成，因此時間響應 會成為分段可微的連續函數，但若能以Lambert W-函數擬合系統響應，則此響應乃由可數無限多個作用在 系統特徵值上的指數函數合成，因此必成為可微函數。本文針對多項延遲的時滯方程，從拉氏變換著手，得 到核函數的拉氏變換，並運用Laurent 級數計算由W-函數組成的核函數，以得到系統響應，這個響應為可 微函數，將可供系統進一步進行穩定性分析與控制設計使用。

For single delay systems, since the state transition function is consecutively obtained by integrating interval by interval and hence the response constructed by the state transition function is only piecewise differentiable. On the other hand, the response of single delay system will be differentiable one if it is constructed via the Lambert W function. In this work, we construct the state transition function by Lambert W function with the aid of Laplace transform together with Laurent series. Afterall the time response function for multiple delay systems is then constructed. This function can be further applied to the stability analysis and controller synthesis.