In the examples involving functions f(t) and/or g(t), we set F(s) = Lff(t)gand TABLE OF LAPLACE TRANSFORM FORMULAS L[tn] = n! 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 … We get the solution y(t) by taking the inverse Laplace transform. What are the steps of solving an ODE by the Laplace transform? – – 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. δ(t ... (and because in the Laplace domain it looks a little like a step function, Γ(s)). 3 2 s t2 (kT)2 ()1 3 2 1 1 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 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! The following table are useful for applying this technique. 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! 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 … 2 1 s t⋅u(t) or t ramp function 4. sn 1 1 ( 1)! 1 δ(t) unit impulse at t = 0 2. s 1 1 or u(t) unit step starting at t = 0 3. Time Domain Function Laplace Domain Name Definition* Function Unit Impulse . Table of Laplace and Z-transforms X(s) x(t) x(kT) or x(k) X(z) 1. Common Laplace Transform Pairs . 12t*e arctan arccot s 16. u(t — 2Tr) sin t 18. s n+1 L−1 1 s = 1 (n−1)! – – Kronecker delta δ0(k) 1 k = 0 0 k ≠ 0 1 2. 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! For example, 4! = n(n 1)(n 2) 3 2 1. = 4 3 2 1 = 24.) 2 1 s t kT ()2 1 1 1 − −z Tz 6. Common Laplace Transform Properties : Name Illustration : Definition of Transform : L st 0: 2 DEFINITION The Laplace transform f (s) of a function f(t) is defined by: ... Laplace Tables.PDF Author: … (sin at) * (cos cot) State the Laplace transforms of a few simple functions from memory. 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. 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 † Deflnition of Laplace transform, † Compute Laplace transform by deflnition, including piecewise continuous functions. 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. 1 − − tn n n = positive integer 5. e as s 1 −
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