Equations Laplace Transform Charts! by MrRail

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									 Math 308: Differential Equations                                      Laplace Transform Charts!

For the entirety of this document, f, g, h are functions that have well defined Laplace trans-
forms, a, b, c are constants, and L{f } is the Laplace transform of f .
Table of Laplace Transforms:
                       f (t)                       F (s) = L{f }(s)

                                                   1
                       1                             ,                s>0
                                                   s
                                                    1
                       eat                             ,              s>a
                                                   s−a
                                                    n!
                       tn , n = 1, 2, 3, ...               ,          s>0
                                                   sn+1
                                                       b
                       sin(bt)                              ,         s>0
                                                   s 2 + b2
                                                         s
                       cos(bt)                                ,       s>0
                                                   s2    + b2
                                                       n!
                       eat tn , n = 1, 2, 3, ...              ,       s>a
                                                   (s − a)n+1

                                                         b
                       eat sin(bt)                               ,    s>a
                                                   (s − a)2 + b2

                                                       s−a
                       eat cos(bt)                               ,    s>a
                                                   (s − a)2 + b2
Note: n! = 1 · 2 · 3...(n − 1)(n). (For example, 1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120, etc.)
Useful Properties:
              L{f + g}       L{f } + L{g}
                               =
                 L{cf }      cL{f }
                               =
              at
           L{e f (t)}(s)     L{f (t)}(s − a)
                               =
               L{f }(s)      sL{f }(s) − f (0)
                               =
              L{f }(s)       s2 L{f }(s) − sf (0) − f (0)
                               =
             L{f (n) }(s)    sn L{f }(s) − sn−1 f (0) − sn−2 f (0) − ... − f (n−1) (0)
                               =
                                d
             L{tf (t)}(s) = − (L{f }(s))
                                ds
                                    dn
            L{tn f (t)}(s) = (−1)n n (L{f }(s))
                                    ds
The unit step function is written:
                                                   0, t < 0
                                     u(t) =
                                                   1, 0 ≤ t
(From Theorem 8) For a > 0, we have:
                         L{f (t − a)u(t − a)}(s) = e−as L{f (t)}(s)

(From Theorem 9) If f has period T and is piecewise continuous on [0, T ], then
                                                               T
                                              1
                             L{f }(s) =                            e−st f (t)dt
                                           1 − e−sT        0


For t > 0, the gamma function Γ(t) is written:
                                                   ∞
                                     Γ(t) =             e−x xt−1
                                               0

It allows us to define the Laplace transform of tr , r > −1:
                                                           Γ(r + 1)
                                     L{tr }(s) =
                                                             sr+1

The convolution of f and g, two piecewise continuous functions defined on [0, ∞) is:
                                                   t
                                (f ∗ g)(t) =           f (t − v)g(v)dv
                                               0

(From Theorems 10 and 11) It satisfies the following:
                                       f ∗g    =        g∗f
                                f ∗ (g + h)    =        (f ∗ g) + (f ∗ h)
                                 (f ∗ g) ∗ h   =        f ∗ (g ∗ h)
                                       f ∗0    =        0
                                  L{f ∗ g}     =        L{f }L{g}

The Dirac delta function δ(t) can be thought of as:
                                  0,                         t=0
                       δ(t) =
                                  ”Something like infinity” , t = 0
It satisfies the following:
                                       ∞
                                           f (t)δ(t)dt = f (0)
                                      −∞
                                    L{δ(t − a)}(s) = e−as

								
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