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2013 | OriginalPaper | Chapter
Table of Laplace Transforms
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Excerpt
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f(t) = ℒ − 1{F(s)}
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F(s) = ℒ{f(t)}
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1
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f (n)(t) (nth derivative)
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s n F(s) − s n − 1 f(0) − ⋯ − f (n − 1)(0)
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2
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f(t − a)H(t − a)
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e − as F(s)
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3
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e at f(t)
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F(s − a)
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4
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(f ∗ g)(t)
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F(s)G(s)
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5
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1
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\(\dfrac{1} {s}\quad (s > 0)\)
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6
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t n (n positive integer)
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\(\dfrac{n!} {{s}^{n+1}}\qquad (s > 0)\)
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7
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e at
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\(\dfrac{1} {s - a}\quad (s > a)\)
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8
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sin(at)
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\(\dfrac{a} {{s}^{2} + {a}^{2}}\quad (s > 0)\)
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9
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cos(at)
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\(\dfrac{s} {{s}^{2} + {a}^{2}}\quad (s > 0)\)
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10
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sinh(at)
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\(\dfrac{a} {{s}^{2} - {a}^{2}}\quad (s > \vert a\vert )\)
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11
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cosh(at)
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\(\dfrac{s} {{s}^{2} - {a}^{2}}\quad (s > \vert a\vert )\)
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12
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δ(t − a) (a ≥ 0)
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e − as
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13
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δ(c(t − a)) (c, a > 0)
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\(\dfrac{1} {c}\,{e}^{-as}\)
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14
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f(t)δ(t − a) (a ≥ 0)
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f(a)e − as
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15
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\(\int \limits _{0}^{t}f(\tau )\,d\tau\)
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\(\dfrac{1} {s}\,F(s)\)
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16
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t n f(t) (n positive integer)
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( − 1) n F (n)(s)
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