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Laplace transform is a transformation that when applied on a given equation makes it easy to integrate. After integrating, one can apply the inverse laplace transform and get the integration of the original gioven integrand.
The Laplace transform is helpful in the solution of ordinary differential describing the behavior of the systems. When the transform operates on a differential equation, "transformed" equation results. It is expressed in terms of an arbitrary complex variable s. The resulting equation is purely algebric and is so easy obtain the result as an explicit result of complex variable...
F(s) = ∫(0,∞) f(t)*exp(-st)dt Laplace Transform
where s = σ+jw
Then you can resume like it:
You use Fourier Series to periodic functions, you use Fourier Transform for non-periodic function ..... but some important non-periodic functions used in control like unit step, ramp, and parabolic functions don't have fourier transforms, because the integral don't converge at infinite .... than Laplace introduce a convergence factor exp(-σt) where σ is a real number that is larger enough to maintain absolute convergence. Look that new transform, is defined between 0 to ∞, because this applies only for time functions, and negative time hasn't physical importance...
hi
s=a+jb in normally.
X(s)=∫x(t)exp(-st).
if we place a=0 and b=ω then it will be fourier transform.
a can be every number but with the limitation of integral it will be defined.
for example when the limitation of integral is -∞ to 0 a must be bigger than 0.
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