Table Of Laplace Transforms Pdf

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Attachment B ii TABLE OF LAPLACE TRANSFORMS f(t) = L−1{F(s)} F(s) = ∞ 0 f(t)e−stdt a, b and c real 1. 1 1 s, Re{s} > 0 2. eat 1 s−a, Re{s} > a 3. tn, n is a positive integer n! sn+1, Re{s} . Table of Laplace Transforms f(x) F(s)=L[f(x)] 1 1 s,s>0 erx 1 s− r,s>r cos βx s s2 +. Table of Laplace Transforms f(t) L{f(t)} =F(s) 1. 1 1 s 2. t 1 s2 3. tn n! sn+1, n apositiveinteger 4. sinkt k s2 +k2 5. coskt s s2 +k2 6. sin2kt 2k2 s(s2 +4k2) 7. cos2kt s2 +2k2 s(s2 +4k2) 8. eat 1 . We'll give two examples of the correct interpretation. First, suppose that f is the constant 1, and has no discontinuity at t = 0. In other words, f is the constant function with value 1. Then we have 0 f = 0, and f(0¡) = 1 (since there is no jump in f at t = 0). Now let's apply the derivative formula above. Table of Laplace Transforms f(t) L(f(t)) f(t) L(f(t)) 1 1 s t 1 s2 Derivatives t2 2 s3 y L(y) tn n! sn+1 y0 sL(y) y(o) eat 1 s a y00 s2L(y) sy(o) y0(0) tneat n! (s a)n+1 cos(!t) s s2 +!2 sin(!t)! s2 +!2 t-Shift cosh(at) s s2 a2 f(t) F(s) sinh(at) a s2 a2 u a(t)f(t a) e asF(s) eat cos(!t) s a (s a)2 +!2 eat sin(!t)! (s a)2 +!2 s-Shift (t a) e as. Table of Laplace transforms f(t) L(f(t)) or F(s) 1. 1 1 s 2. eat 1 s−a 3. tn n! sn+1 n≥0 integer 4. eattn n! (s−a)n+1 n≥0 integer 5. sinkt k s2 +k2 6. coskt s s2 +k2 7. eatsinkt k (s−a)2 +k2 8. eatcoskt s−a (s−a)2 +k2 9. 1 √ t r π s u(t−a) e−as s a≥0 δ(t−a) e−as a≥0. Table of Laplace Transforms f(t) L{f(t)} =F(s) 1. 1 1 s 2. t 1 s2 3. tn n! sn+1, n apositiveinteger 4. sinkt k s2 +k2 5. coskt s s2 +k2 6. sin2kt 2k2 s(s2 +4k2) 7. cos2kt s2 +2k2 s(s2 +4k2) 8. eat 1 s−a 9. sinhkt k s2 −k2 coshkt s s2 −k2 eatt 1 (s−a)2 f(t) L{f(t)} =F(s) eattn n! (s−a)n+1, n apositiveinteger eatsinkt k. TABLE OF LAPLACE TRANSFORMS f(t) = L−1{F(s)} F(s) = ∞ 0 f(t)e−stdt a, b and c real 1. 1 1 s, Re{s} > 0 2. eat 1 s−a, Re{s} > a 3. tn, n is a positive integer n! sn+1, Re{s} > 0 4. tp,p > −1 Γ(p+1) sp+1, Re{s} > 0 5. sin(at) a s2+a2, Re{s} > 0 6. cos(at) s s2+a2, Re{s} > 0 7. sinh(at) a s2−a2, Re{s} > |a| 8. cosh(at) s s2−a2, Re{s. State the Laplace transforms of a few simple functions from memory. What are the steps of solving an ODE by the Laplace transform? In what cases of solving ODEs is the present method preferable to that in Chap. 2? What property of the Laplace transform is crucial in solving ODEs? = Explain. When and how do you use the unit step function and.