LAPLACE TRANSFORM DEPARTMENT OF MATHEMATICS NIT Calicut MA1001 MATHEMATICS I Krishnan Paramasivam

Simon D Poisson Pierre-Simon Laplace 1749 - 1827 (French Mathematician) Developed mathematics in astronomy, physics, and statistics. Began work in calculus which led to the Laplace Transform. Focused later on celestial mechanics. One of the first scientists to suggest the existence of “black holes” in space. Heart of Probability Theory: “Central Limit Theorem” (1810) – tool to solve of method of least squares. Laplace

Black Hole - region of space in which the gravitational field is so powerful. Black hole is one-way surface. Objects can fall, but out of which nothing can come. It is “Black" because it absorbs all light that hits, reflect nothing.

History of the Transform Euler began looking at integrals as solutions to DE in mid 1700s. Lagrange took this a step further while working on probability density functions and looked at forms of the following equation: Finally, in 1785, Laplace began using a transformation to solve equations of finite differences which eventually lead to the current transform

TRANSFORM D cosx sinx x2 2x cosx+2x sinx+x2 D - LINEAR OPERATOR D(sinx) =cosx set of all differentiable functions D cosx sinx x2 2x cosx+2x sinx+x2 D - LINEAR OPERATOR D(sinx+x2) = cosx+2x = D(sinx)+D(x2)

Differential Transform Integral Transform

Integral Transform - LAPLACE TRANSFORM DEFINITION ) assume that integral exists function of s

PROPERTIES OF LAPLACE TRANSFORM RESULT 1 Let

n=0 n=1 n=2

L is linear integral operator

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8. (by definition) (by definition)

This section ends… we move on to see few illustrative examples… (above slides may have few typing errors…please verify with your classnotes)