𝑓(𝑡) 𝑡

Apuntes

𝑓(𝑡) 𝑡 ≥ 0 𝑓(𝑡) ℒ 𝑓 𝑡 = න 0 ∞ 𝑒−𝑠𝑡𝑓 𝑡 𝑑𝑡 𝑠 ℒ 𝑓 𝑡 𝑠 𝐹 𝑠 = ℒ 𝑓 𝑡 ℒ 1 ℒ 𝑡 ℒ 𝑡𝑛 , 𝑛 ∈ ℕ ℒ 𝑡𝑛 = න 0 ∞ 𝑒−𝑠𝑡𝑡𝑛𝑑𝑥 = − 1 𝑠 ∙ ቤ 𝑡𝑛 𝑒𝑠𝑡 0 ∞ + 𝑛 𝑠 න 0 ∞ 𝑒−𝑠𝑡𝑡𝑛−1𝑑𝑡 = 𝑛 𝑠 ℒ 𝑡𝑛−1 ℒ 𝑡𝑛 = 𝑛 𝑠 ℒ 𝑡𝑛−1 ℒ 𝑡𝑛 = 𝑛 𝑠 ∙ 𝑛 − 1 𝑠 ℒ 𝑡𝑛−2 ℒ 𝑡𝑛 = 𝑛 𝑠 ∙ 𝑛 − 1 𝑠 ∙ 𝑛 − 2 𝑠 ℒ 𝑡𝑛−3 ⋮ ℒ 𝑡𝑛 = 𝑛 𝑠 ∙ 𝑛 − 1 𝑠 ∙ 𝑛 − 2 𝑠 ∙ ⋯ ∙ 3 𝑠 ∙ 2 𝑠 ∙ 1 𝑠 ℒ 1 ℒ 𝑡𝑛 = 𝑛(𝑛 − 1)(𝑛 − 2) ⋯ 3 ∙ 2 ∙ 1 𝑠𝑛 ∙ 1 𝑠 ℒ 𝑡𝑛 = 𝑛! 𝑠𝑛+1 ℒ 𝑒𝑎𝑡 𝛼, 𝛽 ∈ ℝ ℒ 𝛼𝑓 𝑡 + 𝛽𝑔(𝑡) = 𝛼ℒ 𝑓 𝑡 + 𝛽ℒ 𝑔 𝑡 ; 𝑠 > 𝑐 𝑐 ℒ 𝑓 𝑡 ℒ 𝑔 𝑡 ℒ 𝛼𝑓 𝑡 + 𝛽𝑔(𝑡) = න 0 ∞ 𝑒−𝑠𝑡 𝛼𝑓 𝑡 + 𝛽𝑔(𝑡) 𝑑𝑡 = 𝛼 න 0 ∞ 𝑒−𝑠𝑡𝑓(𝑡)𝑑𝑡 + 𝛽 න 0 ∞ 𝑒−𝑠𝑡𝑔(𝑡)𝑑𝑡 = 𝛼ℒ 𝑓 𝑡 + 𝛽ℒ 𝑔 𝑡 ∎ ℒ cos 𝑎𝑡 ℒ sin 𝑎𝑡 ℒ cosh 𝑎𝑡 ℒ sinh 𝑎𝑡 cosh(𝑥) sinh(𝑥) cosh 𝑥 = 𝑒𝑥 + 𝑒−𝑥 2 sinh 𝑥 = 𝑒𝑥 − 𝑒−𝑥 2 ℒ 4 + 5 sin 2𝑡 − 3𝑡4 + 8cos(3𝑡) ℒ 𝑡𝑛 = 𝑛! 𝑠𝑛+1 ℒ s𝑖𝑛ℎ(𝑎𝑡) = 𝑎 𝑠2 − 𝑎2 ℒ 𝑒𝑎𝑡 = 1 𝑠 − 𝑎 ℒ cosℎ(𝑎𝑡) = 𝑠 𝑠2 − 𝑎2 ℒ cos(𝑎𝑡) = 𝑠 𝑠2 + 𝑎2 ℒ sin(𝑎𝑡) = 𝑎 𝑠2 + 𝑎2 ℒ 𝑓 𝑡 = ׬0 ∞ 𝑒−𝑠𝑡𝑓 𝑡 𝑑𝑡 ℒ 1 𝑡 ℒ 𝑒𝑡2 𝑓(𝑡) ℒ 𝑓 𝑡 𝑓(𝑡) [0, ∞) 0 ≤ 𝑎 ≤ 𝑡 ≤ 𝑏 𝑡1, 𝑡2, … , 𝑡𝑛 𝑓 𝑡𝑘−1, 𝑡𝑘 , 𝑘 = 2, … , 𝑛 𝑓(𝑡) 𝑐 𝑐, 𝑀, 𝑇 > 0 𝑓 𝑡 ≤ 𝑀𝑒𝑐𝑡 ; ...