Laplace Foundation Transformations.

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23 Terms

1
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f(t)

L{f(t)} = F(s)

2
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1

1/s

3
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t

1/s²

4
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t^n

n! / s^(n+1)

5
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e^(at)

1 / (s - a)

6
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sin(at)

a / (s² + a²)

7
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cos(at)

s / (s² + a²)

8
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sinh(at)

a / (s² - a²)

9
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cosh(at)

s / (s² - a²)

10
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u(t - a)

f(t - a)u(t - a) → e^(-as)F(s)

11
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e^(at)f(t)

F(s - a)

12
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δ(t - a)

e^(-as)

13
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t * sin(at)

2as / (s² + a²)²

14
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t * cos(at)

(s² - a²) / (s² + a²)²

15
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t * e^(at)

1 / (s - a)²

16
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f'(t)

sF(s) - f(0)

17
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f''(t)

s²F(s) - s f(0) - f'(0)

18
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integral of f(t) from 0 to t

F(s)/s

19
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heaviside step at t=a

e^(-as)/s

20
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ramp function (t*u(t))

1/s²

21
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unit impulse δ(t)

1

22
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rectangular pulse (u(t) - u(t - T))

(1 - e^(-sT))/s

23
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exponential decay e^(-at)

1 / (s + a)