MMfE - Examples of Laplace Transform

All Exercises (with Solutions) containing the “Laplace_transform” tag:

• MMfE ~ exam_2021_02_17 - with solutions
• MMfE ~ exam_2021_06_25 - with solutions
• MMfE ~ exam_2021_09_10 - with solutions

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General Solution

Lets start by answering the question:

• “Is $f(t)$ Laplace Transformable?”

If

$$\exists \alpha \in \mathbb{R},M>0:|f(t)|\le M{e}^{\alpha t}\text{for}t>0$$

→ $F(s)=\mathcal{L}[f(t)]$ is well-defined for all complex values of $s$ such that: $\mathrm{Re}\{s\}>\alpha$.

More Reading: MMfE - Is Laplace Transformable

• The Laplace Transform is given by:

$$f(t)={\int }_0^{\infty }f(t)e^{-st}dt$$

Where $\sigma$ is the abscissa of convergence

• The Inverse Laplace Transform is given by:

$$f(t)={\int }_{\sigma -i\infty }^{\sigma +i\infty }F(s)e^{st}ds$$

Where $\sigma$ is the abscissa of convergence

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MMfE - Cheatsheet 'Laplace Transform'

Cheat-Sheet

Link to original

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~Ex.: 2021_02_17 Point 4.

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~Ex.: 2021_06_25 Point 3.

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~Ex.: 2021_06_25 Point 4.

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~Ex.: 2021_09_10 Point 3.

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~Ex.: 2021_09_10 Point 4.

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