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Additional info for Schaum's Outline of Laplace Transforms
Example text
3. s i n t s t' n = 0, 1, 2, ... eat 1 4. s-a 1 sin at 82 + a2 a 5. cos at 6. 82 + a2 7 1 sinh at 82 - a2 a S 8. s2 cosh at a2 SOME IMPORTANT PROPERTIES OF INVERSE LAPLACE TRANSFORMS In the following list we have indicated various important properties of inverse Laplace transforms. Note the analogy of Properties 1-8 with the corresponding properties on Pages 3-5. V 1. Linearity property. Theorem 2-2. If c1 and C2 are any constants while f l (s) and f2(s) are the Laplace transforms of Fl (t) and F2(t) respectively, then {Clfl(s) + C2f2(s)} = C1'-1 (f1(s)) + C2 °l 1 (f2 (S)) (1) ciFi(t) + c2F2(t) The result is easily extended to more than two functions.
2 e 464 (t - 3)3/2 e-4(t-3) t>3 3V t<3 0 4 (t - 3)3/2 e-4(t-4) t>3 3V-ir t<3 0 4 (t - 3)3/2 a-4(t-4) 3' 'u(t - 3) 10. Prove the change of scale property: If C-' (f(s)) = F(t), then i-1 {f(ks)) = Method 1. By Problem 11, Page 14, we have on replacing a by 1/k, C {F(t/k)} = k f(ks). e 1{f(ks)} Method 2. C-1 { f(ks)} = k F(t/k). 1/251 =cos 11. If s t I find -1/ks i l(ks)1/2 e 1/2$ s where a > 0. 1 c o s 2 -t/-k- cos 2 k 7r(tlk) Vi/-k &t or e-1/kst 81/2 I cos 2'/7 k V;L Then letting k = 1/a, 7 1 °e Je 81//2 l at cos 2 2 Tf t INVERSE LAPLACE TRANSFORMS OF DERIVATIVES AND INTEGRALS 12.
U so that F'(t) = Taking the Laplace transform, we have - ds Is f (s) - F(0)} = Then by integration, _ s f(s) or s2 4- 1 ds 82+ 1 4- In (82 + 1) + c By the final-value theorem, lim s f (s) = lim F(t) = 0 so that c = 0. t-. 0 S IS f(s)} f(s) = or s f(s) _ 4- In (S2 + 1) Thus In (82 + 1) 2s We can also use Method 4 of Problem 36 [see Problem 153]. 38. e {Ei (t)} Let F(t) f ey° du. u Is f ( s ) Integrating, fa euu = In (s+ 1) du } J Then tF'(t) = -e-t. - F(0)} = $+I s f(s) Taking the Laplace transform, we find or ds IS f (s)} = 8+1 In (s + 1) + c = Applying the final-value theorem as in Problem 37, we find c = 0 and so f(s) = In (s + 1) s For another method similar to that of Method 4, Problem 36, see Problem 153.