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trojansc82
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Homework Statement
Laplace Transform of e-t sin t
Homework Equations
The Attempt at a Solution
I have the solution, but I am unable to figure out how the denominator becomes 1/[(s + 1)2 + 1]
rock.freak667 said:The presence of the eat would case the shift from 's' to 's-a'. This is why it is called the shift theorem, it's mainly used in the inverse laplace transform.
So you know that L{sint} = 1/(s2+1)
and following shift theorem L(eatsint) = 1/[(s-a)2+1].
You can derive it too using the integral formula.
trojansc82 said:I am unable to derive it from the integral formula. I need to see the steps. I'm fairly certain I've been able to integrate it correctly, but I keep getting a repetitive e^-t sin t or e^-t cos t when I integrate.
The Laplace Transform is a mathematical tool used to transform a function of time into a function of complex frequency. It is commonly used in engineering and physics to simplify differential equations and solve problems involving time-dependent systems.
While both transforms involve changing a function from one domain to another, the Laplace Transform is used for functions that are time-dependent and the Fourier Transform is used for functions that are periodic. Additionally, the Laplace Transform can be used for functions that do not have a Fourier Transform.
The Laplace Transform is calculated by taking the integral of a function multiplied by the exponential function e^-st, where s is a complex number. This integral is known as the Laplace Transform integral and can be solved using various methods, such as partial fraction decomposition or the use of a table of common Laplace Transform pairs.
The Laplace Transform has many applications in various fields, including electrical engineering, control systems, signal processing, and quantum mechanics. It is commonly used to model and analyze time-dependent systems, such as electronic circuits, mechanical systems, and chemical reactions.
While the Laplace Transform is a powerful tool, it does have some limitations. One limitation is that the function being transformed must be defined for all positive time values. Additionally, the Laplace Transform is not always unique, meaning that multiple functions can have the same Laplace Transform. This can make it challenging to find the original function from its Laplace Transform.