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Laplace Transform of Some Basic Functions

  1. Exponential Function
Description: 4929.png
By definition of Laplace transform,
Description: 4935.png
Similarly, for f(t) = eatDescription: 4952.png
  1. Unit Step Function or, Heaviside Unit Function
f(t) = u(t) = 1 for t > 0
= 0 for t < 0
and is undefined for t = 0.
Description: Description: 1103.png
(a) Unit step function
Description: 4972.png
Description: Description: 1140.png
(b) Shifted unit step function
Also, the Laplace transform of step function of magnitude K is
Description: 4978.png 
Similarly, the Laplace transform of the shifted unit step function u(t – T) is,
Description: 4984.png {by differentiation theorem}
Another function, called gate function can be obtained from step function as follows.
Description: 2057.png
Gate function
Therefore, g(tDescription: 4990.png and, Description: 4999.png
  1. The Sine Function
f (t) = Description: 5005.png
F(s) = Description: 5011.pngDescription: 5017.png
Description: 5024.png
  1. The Cosine Function
f (t) = Description: 5030.png
F(s) = Description: 5036.png
Description: 5042.png
  1. The Hyperbolic Sine Function
f (t) = Description: 5049.png
F(s) = Description: 5069.png
Description: 5075.png
  1. The Hyperbolic Cosine Function
f (t) = Description: 5082.png
F(s) = Description: 5091.png
Description: 5099.png
  1. The Damped Sinusoidal Function
f (t) = Description: 5105.png
F(s) = Description: 5111.png
Description: 5117.png
Description: 5123.png
  1. The Damped Cosine Function
f (t) = Description: 5129.png
F(s) = Description: 5135.png
Description: 5141.png 
Description: 5147.png
  1. The Ramp Function
Description: Description: 1196.png
Ramp function
Description: 5153.png
Description: 5168.png
Integrating by parts, let,
u = tn and dv = e– st dt
then Description: 5175.png and Description: 5181.png
Now, F(s) = Description: 5187.png
Description: 5193.png
Description: 5199.png
Description: 5210.png
For n =1, Description: 5220.png
For n =2, Description: 5229.png
  1. Impulse Function or Dirac Delta Function [δ (t)]
Description: Description: 1184.png
Generation of impulse function from gate function
It is a function of a real variable t, such that the function is zero everywhere except at the instant t = 0. Physically, it is a very sharp pulse of infinitesimally small width and very large magnitude, the area under the curve being unity.
Consider a gate function as shown in figure.
The function is compressed along the time-axis and stretched along the y-axis, keeping area under the pulse unity. As a0, the value of Description: 5235.png and the resulting function is known as impulse.
It is defined as, δ (t) = 0 for t  0
and Description: 5241.png
Also, δ (t) = Description: 5247.png
The Laplace transform of the impulse function is obtained as,
[δ (t)] Description: 5253.png
Description: 5259.png
Description: 5267.png
Description: 5285.png
= 1

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