# finalCrib - ENGI 5821 — FORMULA SHEET FOR FINAL EXAM...

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Unformatted text preview: ENGI 5821 — FORMULA SHEET FOR FINAL EXAM Modeling Laplace Transform: L{ f ( t ) } = F ( s ) = Z ∞- f ( t ) e- st dt The Inverse Laplace Transform: L- 1 { F ( s ) } = 1 2 πj Z σ + j ∞ σ- j ∞ F ( s ) e st ds = f ( t ) u ( t ) The Laplace Transform Pair Table: f ( t ) F ( s ) 1. δ ( t ) 1 2. u ( t ) 1 s 3. tu ( t ) 1 s 2 4. t n u ( t ) n ! s n +1 5. e- at u ( t ) 1 s + a 6. (sin ωt ) u ( t ) ω s 2 + ω 2 7. (cos ωt ) u ( t ) s s 2 + ω 2 Laplace Transform Theorems: Theorem Name 1. L{ f ( t ) } = F ( s ) = R ∞- f ( t ) e- st dt Definition 2. L{ kf ( t ) } = kF ( s ) Linearity theorem 3. L{ f 1 ( t ) + f 2 ( t ) } = F 1 ( s ) + F 2 ( s ) Linearity theorem 4. L{ e- at f ( t ) } = F ( s + a ) Frequency shift theorem 5. L{ f ( t- T ) } = e- sT F ( s ) Time shift theorem 6. L{ f ( at ) } = 1 a F ( s a ) Scaling theorem 7. L{ df dt } = sF ( s )- f (0- ) Differentiation 8. L{ d 2 f dt 2 } = s 2 F ( s )- sf (0- )- f (0- ) Differentiation 9. L{ d n f dt n } = s n F ( s )- ∑ n k =1...
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finalCrib - ENGI 5821 — FORMULA SHEET FOR FINAL EXAM...

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