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Turion
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Please ignore this thread.
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brmath said:It would have helped if you stated Theorem 4.3.3. That said, we could try it without any special theorem. We have
##Lf = \int_0^{\infty} e^{-st}f(t)dt ##
which you can split up into the sum of integrals from (0,a) (a,2a) ... ([n-1]a,na) (na,t), where na < t < (n+1)a.
Each of these is easy enough to evaluate, but be careful to separate out the case where n is even and n is odd (why?). I'll bet you can factor some stuff out of the resulting sum.
Why don't you try this out? And why don't you quote theorem 4.3.3 -- maybe it's an easier way to do things.
Actually, I think the solutions manual is wrong. Please ignore this thread.
A Laplace Transform is a mathematical tool used to convert a function of time into a function of complex frequency. It is commonly used in engineering and physics to solve differential equations.
The Laplace Transform allows us to solve complex differential equations that cannot be easily solved using other methods. It also helps to simplify calculations and make them more manageable.
The Laplace Transform is calculated by taking the integral of a function multiplied by an exponential term. This integral can be evaluated using tables, software, or by hand using integration techniques.
The Laplace Transform is commonly used in electrical engineering, control systems, signal processing, and other fields to model and analyze systems. It is also used in physics to solve problems involving differential equations.
The Laplace Transform is not always applicable, especially for functions that do not have a finite integral. It also only works for linear, time-invariant systems, and cannot be used for nonlinear or time-varying systems. Additionally, the inverse Laplace Transform may not exist for all functions, making it difficult to obtain the original function from its Laplace Transform.