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The Laplace Transform of step functions (Sect. 6.3). ◮ Overview and notation. ◮ The deﬁnition of a step function. ◮ Piecewise discontinuous functions. ◮ The Laplace Transform of discontinuous functions. ◮ Properties of the Laplace Transform.

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Notation: −1 If L[f (t)] = F(s), then we denote L [F (s)] = f (t). Remark: One can show that for a particular type of functions f , that includes all functions we work with in this Section, the notation above is well-deﬁned. Example [ ] 1 at From the Laplace Transform table we know that L e = . s − a [ ] 1 −1 at Then also holds that L = e . ⊳ s − a Overview and notation. Overview: The Laplace Transform method can be used to solve constant coeﬃcients diﬀerential equations with discontinuous source functions.

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Remark: One can show that for a particular type of functions f , that includes all functions we work with in this Section, the notation above is well-deﬁned. Example [ ] 1 at From the Laplace Transform table we know that L e = . s − a [ ] 1 −1 at Then also holds that L = e . ⊳ s − a Overview and notation. Overview: The Laplace Transform method can be used to solve constant coeﬃcients diﬀerential equations with discontinuous source functions. Notation: −1 If L[f (t)] = F(s), then we denote L [F (s)] = f (t).

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Example [ ] 1 at From the Laplace Transform table we know that L e = . s − a [ ] 1 −1 at Then also holds that L = e . ⊳ s − a Overview and notation. Overview: The Laplace Transform method can be used to solve constant coeﬃcients diﬀerential equations with discontinuous source functions. Notation: −1 If L[f (t)] = F(s), then we denote L [F (s)] = f (t). Remark: One can show that for a particular type of functions f , that includes all functions we work with in this Section, the notation above is well-deﬁned.

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[ ] 1 −1 at Then also holds that L = e . ⊳ s − a Overview and notation. Overview: The Laplace Transform method can be used to solve constant coeﬃcients diﬀerential equations with discontinuous source functions. Notation: −1 If L[f (t)] = F(s), then we denote L [F (s)] = f (t). Remark: One can show that for a particular type of functions f , that includes all functions we work with in this Section, the notation above is well-deﬁned. Example [ ] 1 at From the Laplace Transform table we know that L e = . s − a

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Overview and notation. Overview: The Laplace Transform method can be used to solve constant coeﬃcients diﬀerential equations with discontinuous source functions. Notation: −1 If L[f (t)] = F(s), then we denote L [F (s)] = f (t). Remark: One can show that for a particular type of functions f , that includes all functions we work with in this Section, the notation above is well-deﬁned. Example [ ] 1 at From the Laplace Transform table we know that L e = . s − a [ ] 1 −1 at Then also holds that L = e . ⊳ s − a

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The Laplace Transform of step functions (Sect. 6.3). ◮ Overview and notation. ◮ The deﬁnition of a step function. ◮ Piecewise discontinuous functions. ◮ The Laplace Transform of discontinuous functions. ◮ Properties of the Laplace Transform.

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Example Graph the step function values u(t) above, and the translations u(t − c) and u(t + c) with c > 0. Solution: ⊳ u(t) u(t + c) The deﬁnition of a step function. Deﬁnition 1 A function u is called a step function at t1= 0 iﬀ holds { 0 for t < 0, u(t) = 1 for t ⩾ 0. u(t - c) 1 - 0c 0 t 0 c t

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Solution: ⊳ u(t) u(t + c) The deﬁnition of a step function. Deﬁnition 1 A function u is called a step function at t1= 0 iﬀ holds { 0 for t < 0, u(t) = 1 for t ⩾ 0. Example Graph the step function values u(t) above, and the translations u(t − c) and u(t + c) with c > 0. u(t - c) 1 - 0c 0 t 0 c t

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⊳ u(t) u(t + c) The deﬁnition of a step function. Deﬁnition 1 A function u is called a step function at t1= 0 iﬀ holds { 0 for t < 0, u(t) = 1 for t ⩾ 0. Example Graph the step function values u(t) above, and the translations u(t − c) and u(t + c) with c > 0. Solution: u(t - c) 1 - 0c 0 t 0 c t

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