Laplace Transform Table PDF
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Paul Dawkins
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This is a table of Laplace transforms, commonly used in engineering and mathematical applications. It shows the relationship between a function of time and its Laplace transform. The table covers various functions and their corresponding transforms.
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Table of Laplace Transforms f (t) = L −1 {F (s)} F (s) = L {f (t)} f (t) = L −1 {F (s)} F (s) = L {f (t)} 1 1 1....
Table of Laplace Transforms f (t) = L −1 {F (s)} F (s) = L {f (t)} f (t) = L −1 {F (s)} F (s) = L {f (t)} 1 1 1. 1 2. ea t s s−a n! Γ (p + 1) 3. tn , n = 1, 2, 3,... 4. tp , p > −1 sn+1 sp+1 √ √ √ π n− 12 1 · 3 · 5 · · · (2n − 1) π 5. t 3 6. t , n = 1, 2, 3,... 1 2s 2 2n sn+ 2 a s 7. sin(at) 8. cos(at) s2 + a2 s2 + a2 2as s2 − a2 9. t sin(at) 10. t cos(at) (s2 + a2 )2 (s2 + a2 )2 2a3 2as2 11. sin(at) − at cos(at) 12. sin(at) + at cos(at) (s2 + a2 )2 (s2 + a2 )2 s s2 − a2 s s2 + 3a2 13. cos(at) − at sin(at) 14. cos(at) + at sin(at) (s2 + a2 )2 (s2 + a2 )2 s sin (b) + a cos (b) s cos (b) − a sin (b) 15. sin(at + b) 16. cos(at + b) s2 + a2 s2 + a2 a s 17. sinh(at) 18. cosh(at) s2 − a2 s2 − a2 b s−a 19. eat sin(bt) 20. eat cos(bt) (s − a)2 + b2 (s − a)2 + b2 b s−a 21. eat sinh(bt) 22. eat cosh(bt) (s − a)2 − b2 (s − a)2 − b2 n! 1 s 23. tn eat , n = 1, 2, 3,... 24. f (ct) F (s − a)n+1 c c e−cs 25. uc (t) = u(t − c) 26. δ(t − c) e−cs s e−cs F (s) e−cs L g(t + c) 27. uc (t)f (t − c) 28. uc (t)g(t) 29. ect f (t) F (s − c) 30. tn f (t), n = 1, 2, 3,... (−1)n F (n) (s) Z ∞ Z t 1 F (s) 31. f (t) F (u) du 32. f (v) dv t s 0 s Z T Z t e−st f (t) dt 0 33. f (t − τ )g(τ ) dτ F (s)G(s) 34. f (t + T ) = f (t) 0 1 − e−sT 35. f 0 (t) sF (s) − f (0) 36. f 00 (t) s2 F (s) − sf (0) − f 0 (0) 37. f (n) (t) sn F (s) − sn−1 f (0) − sn−2 f 0 (0) · · · − sf (n−2) (0) − f (n−1) (0) © Paul Dawkins - https://tutorial.math.lamar.edu
