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Table of Laplace Transforms
Table Notes
- This list is not a complete listing of Laplace transforms and only contains some of the more commonly used Laplace transforms and formulas.
- Recall the definition of hyperbolic functions.
- Be careful when using “normal” trig function vs. hyperbolic functions. The only difference in the formulas is the “+a2” for the “normal” trig functions becomes a “−a2” for the hyperbolic functions!
- Formula #4 uses the Gamma function which is defined as
If n is a positive integer then,
Γ(n+1)=n!
The Gamma function is an extension of the normal factorial function. Here are a couple of quick facts for the Gamma function
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FAQs on Table of Laplace Transforms - Calculus - Mathematics
| 1. What is a Laplace transform? |  |
Ans. A Laplace transform is a mathematical operation that transforms a function of time, often a signal, into a function of a complex variable s, which represents frequency in the Laplace domain. It is commonly used in engineering and physics to solve differential equations and analyze linear systems.
| 2. How is the Laplace transform defined? | |
Ans. The Laplace transform of a function f(t), denoted by F(s), is defined as the integral of f(t) multiplied by e^(-st), where s is a complex number. The integral is taken from 0 to infinity. Mathematically, it can be represented as F(s) = ∫[0,∞] f(t) e^(-st) dt.
| 3. What are the advantages of using Laplace transforms? | |
Ans. There are several advantages of using Laplace transforms. Firstly, they can simplify the solution of differential equations by transforming them into algebraic equations. This makes it easier to solve complex problems. Secondly, Laplace transforms allow for the analysis of systems in the frequency domain, providing insights into system behavior at different frequencies. Finally, they have applications in control systems, circuit analysis, signal processing, and many other fields of engineering.
| 4. Can Laplace transforms be used to solve non-linear equations? | |
Ans. Laplace transforms are primarily used for linear systems and linear differential equations. However, they can sometimes be used to solve non-linear equations by linearizing them around a specific operating point. This involves approximating the non-linear function as a linear function in the vicinity of the operating point and then applying the Laplace transform. However, this approach may introduce errors and is not always applicable to all non-linear equations.
| 5. What is the inverse Laplace transform? | |
Ans. The inverse Laplace transform is the operation that converts a function F(s) in the Laplace domain back to its corresponding function f(t) in the time domain. It is denoted by L^(-1){F(s)}. The inverse Laplace transform is typically found using tables of Laplace transform pairs or by applying partial fraction decomposition and the residue theorem. It allows us to obtain the original function from its Laplace transform, enabling the solution of differential equations and the analysis of system responses in the time domain.
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