Laplace Transformation Table

Jordan Emanuel

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08-12-2024

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The Laplace transform is a powerful mathematical tool used extensively in engineering and physics, particularly in solving linear differential equations. This table provides a concise summary of common Laplace transforms, useful for students and professionals alike. Remember that the Laplace transform of a function f(t) is denoted as ℒ{f(t)} or F(s).

Basic Functions and Their Laplace Transforms

Important Properties of Laplace Transforms

Understanding the properties below is crucial for efficiently applying Laplace transforms to solve complex problems.

• Linearity: ℒ{af(t) + bg(t)} = aℒ{f(t)} + bℒ{g(t)} where 'a' and 'b' are constants.
• Time Shifting: If ℒ{f(t)} = F(s), then ℒ{f(t-a)u(t-a)} = e^-asF(s), where u(t-a) is the unit step function shifted by 'a'.
• Frequency Shifting: If ℒ{f(t)} = F(s), then ℒ{e^atf(t)} = F(s-a).
• Differentiation in the Time Domain: ℒ{f'(t)} = sF(s) - f(0) and ℒ{f''(t)} = s²F(s) - sf(0) - f'(0).
• Integration in the Time Domain: ℒ{∫₀^tf(τ)dτ} = F(s)/s

Beyond the Basics

This table represents a subset of commonly encountered Laplace transforms. More extensive tables can be found in engineering mathematics textbooks and online resources. Remember to always consider the conditions for convergence when using these transforms. Careful application of the properties listed above allows for the solution of many complex differential equations that would be difficult or impossible to solve using other methods. Mastering the Laplace transform is a valuable asset for anyone working in fields that rely on mathematical modeling and analysis.