(sin at) * (cos cot) State the Laplace transforms of a few simple functions from memory. = n(n 1)(n 2) 3 2 1. In the examples involving functions f(t) and/or g(t), we set F(s) = Lff(t)gand 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! 12t*e arctan arccot s 16. u(t — 2Tr) sin t 18. 2 1 s t⋅u(t) or t ramp function 4. sn 1 1 ( 1)! 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! s n+1 L−1 1 s = 1 (n−1)! δ(t ... (and because in the Laplace domain it looks a little like a step function, Γ(s)). Table of Laplace and Z-transforms X(s) x(t) x(kT) or x(k) X(z) 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 … 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! Time Domain Function Laplace Domain Name Definition* Function Unit Impulse . = 4 3 2 1 = 24.) 2 1 s t kT ()2 1 1 1 − −z Tz 6. TABLE OF LAPLACE TRANSFORMS f(t) 1. 2 DEFINITION The Laplace transform f (s) of a function f(t) is defined by: ... Laplace Tables.PDF Author: … The following table are useful for applying this technique. 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 TRANSFORM FORMULAS L[tn] = n! Common Laplace Transform Properties : Name Illustration : Definition of Transform : L st 0: 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. LAPLACE TRANSFORM TABLES MATHEMATICS CENTRE ª2000. – – Kronecker delta δ0(k) 1 k = 0 0 k ≠ 0 1 2. 1 δ(t) unit impulse at t = 0 2. s 1 1 or u(t) unit step starting at t = 0 3. 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 … 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! † Deflnition of Laplace transform, † Compute Laplace transform by deflnition, including piecewise continuous functions. For example, 4! 1 − − tn n n = positive integer 5. e as s 1 − Common Laplace Transform Pairs . 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 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. We get the solution y(t) by taking the inverse Laplace transform.