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### Laplace Transform

• The idea of Laplace transform is to solve an equation containing differential and
integral terms by transforming the equation into the S-space
• This is done by applying the Laplace transforming function as follows:- { f(t)} = • The resulting expression is a function of s which we say F(s)
• Normally, people do not have to solve the entire formula as we can use standard
Laplace transformations. Below is the table :-

 Time Function f(t)    f(t) =  -1{F(s)} Laplace Transform of f(t) F(s) =  { f(t)} F1  s > 0 t (unit-ramp function)  s > 0 tn (n, a positive integer)  s > 0 eat  s > a sin ωt  s > 0 cos ωt  s > 0 tng(t), for n = 1, 2, ...  t sin ωt  s > |ω| t cos ωt  s > |ω| g(at)  Scale property eatg(t) G(s − a)       Shift property eattn, for n = 1, 2, ...  s > a te-t  s > -1 1 − e-t/T  s > -1/T eatsin ωt  s > a eatcos ωt  s > a u(t)  s > 0 u(t − a)  s > 0 u(t − a)g(t − a) e-asG(s) Time-displacement theorem g'(t) sG(s) − g(0) g''(t) s2 • G(s) − s • g(0) − g'(0) g(n)(t) sn • G(s) − sn-1 • g(0) − sn-2 • g'(0) − ... − g(n-1)(0)      • The last 5 on the table shows the true power of Laplace transform. A complex
integral calculation then would be impossible to be performed in the time
domain, when transformed into the S domain can be easily represented
• Laplace transforms have several properties which are:-

·        Property1 - Constant Multiple

·        If
a is a constant and f(t) is a function of t, then L{a f(t)} = a L{f(t)}

·        Example
{7 sin t} = 7{sin t}

·        Property2 - Linearity Property

·        If
a and b are constants while f(t) and g(t) are functions of t, then

L{a f(t) + b g(t)} = a L{f(t)}
+ b L{g(t)}

·        Example
L{3t + 6t2 } = 3 L{t} + 6 L{t2}

·        Property3. Change of Scale Property

·        If
L{f(t)} = F(s) then L{f(at)} = 1/a F(s/a)

·        Example
L{F(5t) = (1/5)F(s/5)

·        Property4. Shifting Property (Shift Theorem)

·        L{exp(at)f(t)}
= F(s − a)

·        Example
L{exp(3t)f(t)} = F(s − 3)

·        Property5. Differential transformation

·        L{tf(t)}
= -F’(s)

·        Property6.

·        The Laplace transforms of the real (or imaginary) part of a complex function is
equal to the real (or imaginary) part of the transform of the complex function.

·        Let Re denote the real part of a complex function C(t) and Im denote the imaginary
part of C(t), then L{Re[C(t)]} = Re L{C(t)} and L{Im[C(t)]} = Im L{C(t)} Open-Plant is a revolutionary Industrial IOT Platform software, used to create and deploy Industrial IT apps/solutions. It is an all-encompassing solution offering both back-end and front-end components i.e. the full stack. From our user's experience, creating and deploying Industrial IT apps became 10x faster and 10x less cost. We serve the mining, energy, oil & gas, construction and manufacturing industry.

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