Young Won Lim1/8/15

Laplace Transform Pairs (4A)

Young Won Lim1/8/15

Copyright (c) 2014 Young W. Lim.

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Pairs (4A) 3 Young Won Lim1/8/15

Selected Laplace Transform Pairs (1)

http://en.wikipedia.org/wiki/Laplace_transform

Pairs (4A) 4 Young Won Lim1/8/15

Selected Laplace Transform Pairs (2)

http://en.wikipedia.org/wiki/Laplace_transform

Pairs (4A) 5 Young Won Lim1/8/15

Unit Step Function

F(s) = ∫0

∞

f (t )⋅e−s t dt

1 u(t) = ∫0

∞

1⋅e−st dt

= limb→∞ [−1

se−st ]

0

b

= limb→∞ [−1

se−sb

+1se−s0]

s > 0 F(s) =1s

1 u(t−a)

1 1

1 1

a

= ∫0

∞

u(t−a)⋅e−st dt

= ∫0

a

u(t−a)⋅e−st dt + ∫a

∞

u(t−a)⋅e−st dt

0<t<a

= ∫a

∞

u(ν)⋅e−s(ν+a) d ν = e−a s∫a

∞

u(ν)⋅e−sν d ν

ν = t−a d ν = dtν+a = t

= e−as⋅1s

F(s) = ∫0

∞

f (t)⋅e−s t dt

1s

1s

1s

e−a s 1s

Pairs (4A) 6 Young Won Lim1/8/15

Unit Step Function

f (t) f (t)u(t)

F (s)

e−a s F(s)

f (t−a) = 1

f (t−a)u(t−a)

a

e−a s F(s)

a

Pairs (4A) 7 Young Won Lim1/8/15

Unit Step Function

F(s) = ∫0

∞

f (t−a)u(t−a)⋅e−st dt

f (t) f (t)u(t)

f (t−a) = 1 f (t−a)u(t−a)

= ∫0

a

f (t−a)u(t−a)⋅e−st dt

+ ∫a

∞

f (t−a)u(t−a)⋅e−s t dt

0<t<a

= ∫a

∞

f (ν)u(ν)⋅e−s (ν+a) d ν

= e−as∫a

∞

f (ν)⋅e−sν d ν

ν = t−ad ν = dt

= e−as⋅F (s)

a

F (s) F (s)

e−a s F(s)e−a s F(s)

Pairs (4A) 8 Young Won Lim1/8/15

Transforms of f(t) and f(t)u(t)

t tu(t )

sin (ω t) sin (ω t)u(t)

cos (ω t ) cos (ω t)u(t)

ωs2

+ω2

s

s2+ω

2

1s

Pairs (4A) 9 Young Won Lim1/8/15

Transforms of (t±1) and (t±1)u(t±1)

t+1 (t+1)u(t+1)

1

s2 −1s

e−s 1

s2

1

s2 +1s

t−1

(t−1)u(t−1)

f 1(t ) = t+1 ≡ f 2(t ) = (t+1)u(t+1) (t ≥ 0)

Pairs (4A) 10 Young Won Lim1/8/15

Translation in the s-domain

e+a t f (t ) F (s − a)

F(s−a) = ∫0

∞

f (t)⋅e−(s−a )t dt = ∫0

∞

[e+a t f (t)]e−s t dt

| 1(s+5) |

arg { 1(s+5) }u(t )

e+5 tu(t)

σ

ω

σ−5

ω

σω

f (s)

Pairs (4A) 11 Young Won Lim1/8/15

cos(ωt)

| s

s2+22 |

arg { s

s2+22 }

cos (ω t ) cos (ω t )u( t)

s

s2+ω

2σ

ω

σω

f (s)

Pairs (4A) 12 Young Won Lim1/8/15

sin(ωt)

| 2

s2+22 |

arg { 2

s2+22 }

sin (ω t) sin (ω t)u(t)

ωs2+ω2σ

ω

σω

f (s)

Pairs (4A) 13 Young Won Lim1/8/15

cosh(ωt)

| s

s2−22 |

arg { s

s2−22 }

cosh (ω t) cosh (ω t)u(t )

s

s2−ω

2σ

ω

σω

f (s)

Pairs (4A) 14 Young Won Lim1/8/15

sinh(ωt)

| 2

s2−22 |

arg { 2

s2−22 }

sinh (ω t) sinh (ω t)u(t)

ωs2−ω2σ

ω

σω

f (s)

Pairs (4A) 15

| s

s2+22 |

arg { s

s2+22 }

| 2

s2+22 |

arg { 2

s2+22 }

| s

s2−22 |

arg { s

s2−22 }

| 2

s2−22 |

arg { 2

s2−22 }

Pairs (4A) 16

| s

s2+22 |

arg { s

s2+22 }

| 2

s2+22 |

arg { 2

s2+22 }

| s

s2−22 |

arg { s

s2−22 }

| 2

s2−22 |

arg { 2

s2−22 }

Pairs (4A) 17 Young Won Lim1/8/15

Plot of es

es = eσ+iω = eσ eiω

|es|= eσ|eiω|= eσ arg { es}= 0 + arg{ eiω

}= ω

σω

f (s)

Pairs (4A) 18 Young Won Lim1/8/15

Translation in the t-domain

f (t−a)u(t−a) e−a s F (s)

∫0

∞

f (t−a)u(t−a)⋅e−st dt

=∫0

a

f (t−a)u(t−a)⋅e−s t dt +∫a

∞

f (t−a)u(t−a)⋅e−s t dt

u(t )

u(t−2)

σ

ω

σ

ω

| e−s

s |

arg { e−s

s }

Young Won Lim1/8/15

References

[1] http://en.wikipedia.org/[2] http://planetmath.org/[3] M.L. Boas, “Mathematical Methods in the Physical Sciences”[4] E. Kreyszig, “Advanced Engineering Mathematics”[5] D. G. Zill, W. S. Wright, “Advanced Engineering Mathematics”[6] T. J. Cavicchi, “Digital Signal Processing”[7] F. Waleffe, Math 321 Notes, UW 2012/12/11[8] J. Nearing, University of Miami[9] http://scipp.ucsc.edu/~haber/ph116A/ComplexFunBranchTheory.pdf