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For example, 4! = n(n 1)(n 2) 3 2 1. 2 DEFINITION The Laplace transform f (s) of a function f(t) is defined by: ... Laplace Tables.PDF Author: … Deflnition: Given a function f(t), t ‚ 0, its Laplace transform F(s) = Lff(t)g is deflned as F(s) = Lff(t)g: = Z 1 0 e¡stf(t)dt = lim: A!1 Z A 0 e¡stf(t)dt We say the transform … 1 δ(t) unit impulse at t = 0 2. s 1 1 or u(t) unit step starting at t = 0 3. Table of Laplace Transforms Definition of Laplace transform 0 L{f (t)} e st f (t)dt f (t) L 1{F(s)} F(s) L{f (t)} Laplace transforms of elementary functions 1 s 1 tn 1! 2 1 s t kT ()2 1 1 1 − −z Tz 6. What are the steps of solving an ODE by the Laplace transform? TABLE OF LAPLACE TRANSFORMS f(t) 1. Laplace transform The bilateral Laplace transform of a function f(t) is the function F(s), defined by: The parameter s is in general complex : Table of common Laplace transform pairs ID Function Time domain Frequency domain Region of convergence for causal systems 1 ideal delay 1a unit impulse 2 delayed nth power with frequency shift We get the solution y(t) by taking the inverse Laplace transform. Table of Laplace Transforms In the table that follows, y(t) is a function of tand Y(s) = Lfy(t)gis the Laplace transform of y(t), where Y(s) := Z 1 0 e sty(t)dt: (Recall that if nis a positive integer, we de ne n! Table 1: Table of Laplace Transforms Number f (t) F (s) 1 δ(t) 2 us(t) 3 t 4 tn 5 e−at 6 te−at 7 1 tn−1e−at (n−1)!81−e−at 9 e−at −e−bt 10 be−bt −ae−at 11 sinat 12 cosat 13 e−at cosbt 14 e−at sinbt 15 1−e−at(cosbt + a b sinbt) 1 1 s 1 s2 n! Laplace Table Page 1 Laplace Transform Table Largely modeled on a table in D’Azzo and Houpis, Linear Control Systems Analysis and Design, 1988 F (s) f (t) 0 ≤ t 1. = 4 3 2 1 = 24.) 1 − − tn n n = positive integer 5. e as s 1 − The following table are useful for applying this technique. 2 1 s t⋅u(t) or t ramp function 4. sn 1 1 ( 1)! tn−1 L eat = 1 s−a L−1 1 s−a = eat L[sinat] = a s 2+a L−1 1 s +a2 = 1 a sinat L[cosat] = s s 2+a L−1 s s 2+a = cosat Differentiation and integration L d dt f(t) = sL[f(t)]−f(0) L d2t dt2 f(t) = s2L[f(t)]−sf(0)−f0(0) L dn … Common Laplace Transform Pairs . Table of Laplace Transforms (continued) a b In t f(t) (y 0.5772) eat) cos cot) cosh at) — sin cot Si(t) 15. et/2u(t - 3) 17. t cos t + sin t 19. † Deflnition of Laplace transform, † Compute Laplace transform by deflnition, including piecewise continuous functions. (sin at) * (cos cot) State the Laplace transforms of a few simple functions from memory. δ(t ... (and because in the Laplace domain it looks a little like a step function, Γ(s)). LAPLACE TRANSFORM TABLES MATHEMATICS CENTRE ª2000. 1 2. t 3. tn na positive integer 4. t1/2 5. t1/2 6. ta 7. sin kt 8. cos kt 9. sin2kt 10. cos2kt 11. eat 12. sinh kt 13. cosh kt 14. sinh2kt 15. cosh2kt 16. teat 17. tneat na positive integer 18. eatsin kt 19. eatcos kt s a (s a)2 k2 k (s a)2 k2 n! 3 2 s t2 (kT)2 ()1 3 2 1 1 – – δ0(n-k) 1 n = k 0 n ≠ k z-k 3. s 1 1(t) 1(k) 1 1 1 −z− 4. s +a 1 e-at e-akT 1 1 1 −e−aT z− 5. Table of Laplace and Z-transforms X(s) x(t) x(kT) or x(k) X(z) 1. Common Laplace Transform Properties : Name Illustration : Definition of Transform : L st 0: In the examples involving functions f(t) and/or g(t), we set F(s) = Lff(t)gand 12t*e arctan arccot s 16. u(t — 2Tr) sin t 18. – – Kronecker delta δ0(k) 1 k = 0 0 k ≠ 0 1 2. Time Domain Function Laplace Domain Name Definition* Function Unit Impulse . TABLE OF LAPLACE TRANSFORM FORMULAS L[tn] = n! s n+1 L−1 1 s = 1 (n−1)!

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