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Laplace Transforms
Speaking about the most used integral transforms in mathematics, you will of course find the Laplace transform being used in most of the areas of this subject. The number of applications of Laplace transforms is outstanding. In addition to mathematics, it has its applications even in the field of probability theory, engineering, economics and physics. The Laplace transform is said to be closely related to another well known transform called Fourier transform. However, in Laplace transform, the moments are resolved out of the functions. On the other hand, the Fourier transform will resolve the functions into different vibration modes. The integral and the differential equations in the field of mathematics can be systematically solved with the help of Laplace transform. Even Fourier transform will help in solving these equations as well.
This transform is employed in the field of engineering and physics for analyzing the optical devices, harmonic oscillators, electrical circuits and invariant systems etc. In almost all these analyses, the Laplace transform will be interpreted to be a time domain transformation, wherein both the outputs as well as the inputs will be the functions with respect to the time.
The man on whom the Laplace transform is named after is Pierre-Simon Laplace. Laplace was the one who used employed the transform in one of his most appreciated works called probability theory. However, since 1744, another well known mathematician called Leonhard Euler carried out his investigations with regard to how integrals were able to find the differential equations solutions. Unfortunately, he was not successful in pursuing this matter for a great extent. Euler was admired by a man called Joseph Lagrange, who later carried out the investigation with regard to the same task. Overall, the Laplace transform was seemed to be very much useful in the field of mathematics, physics and engineering.
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