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Practicing more about laplace and inverse laplace transform with their application to real life
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LAPLACE AND INVERSE LAPLACE TRANSFORM AND THEIR APPLICATIONSINTRODUCTION
In mathematics, The Laplace transform (or Laplace method) is named in honor of the
great French mathematician Pierre Simon De Laplace (1749-1827). Systematically developed
by the British physicist Oliver Heaviside (1850–1925), to simplify the solution of
many differential equations that describe physical processes. Today it is used most frequently
by electrical engineers in the solution of various electronic circuit problems.
This method is used to find the approximate value of the integration of the given function.
Laplace transform changes one signal into another according to some fixed set of rules or
equations. The Laplace transform is an integral transform that converts a function of real
variable to a complex variable i.e. Time domain to Frequency domain
The transform has many applications in science and engineering because it is a tool for
solving differential equation. In particular, it transforms ordinary differential equations into
algebraic equations and convolution into multiplication.
Page 1
, LAPLACE AND INVERSE LAPLACE TRANSFORM AND THEIR APPLICATIONS
CHAPTER 1
LAPLACE AND INVERSE LAPLACE TRANSFORM
1.1 DEFINITION
Let f(t), be a function defined for all positive real values of t, and s be real or complex
parameters, then Laplace transform of f(t), is denoted by L[f(t)] or F(s) and is defined as
∞
𝐿[𝑓(𝑡)] = ∫0 𝑒 −𝑠𝑡 𝑓 (𝑡)𝑑𝑡, provided integral exists
1.2 TRANSFORM OF SOME STANDARD FUNCTIONS
1. Laplace transform of constant
∞
Proof: We have 𝐿[𝑓(𝑡)] = ∫0 𝑒 −𝑠𝑡 𝑓 (𝑡)𝑑𝑡
∞
Therefore, 𝐿[𝑎] = ∫0 𝑒 −𝑠𝑡 𝑎 𝑑𝑡
∞
= 𝑎 ∫0 𝑒 −𝑠𝑡 𝑑𝑡
∞
𝑒 −𝑠𝑡 𝑎
=𝑎 ] = − [ 0 − 1]
−𝑠 0 𝑠
𝒂
𝑳[𝒂] = 𝒔
2. A. Laplace transform of 𝒆−𝒂𝒕
∞
Proof: We have 𝐿[𝑓(𝑡)] = ∫0 𝑒 −𝑠𝑡 𝑓 (𝑡)𝑑𝑡
∞ ∞
Therefore, 𝐿[𝑒 −𝑎𝑡 ] = ∫0 𝑒 −𝑠𝑡 𝑒 −𝑎𝑡 𝑑𝑡 = ∫0 𝑒 −𝑠𝑡−𝑎𝑡 𝑑𝑡
∞
= ∫0 𝑒 −(𝑠+𝑎)𝑡 𝑑𝑡
∞
𝑒 −(𝑠−𝑎)𝑡 1
= ] = − [ 0 − 1]
−(𝑠−𝑎) 0 𝑠−1
1
𝐿[𝑒 −𝑎𝑡 ] = 𝑠−1
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