LAPLACE TRANSFORM
=================
          /<inf>
          |
L[f(t)] = | e^(-st) f(t) dt
          |
          /0





Laplace Transform Pairs
-----------------------

    f(t)            F(s)            Name
-----------------------------------------------------
<sigma>(t)          1               unit impulse
H[t]                1/s             Heavyside step function
e^(-at)             1/(s+a)
t                   1/(s^2)
(t^n)/n!            1/(s^(n+1))
t(e^(-at))          1/((s+a)^2)
(t^n)(e(-at))/(n!)  1/((s+a)^(n+1))
sin(bt)             b/((s^2)+(b^2))
cos(bt)             s/((s^2)+(b^2))
e^(-at)sin(bt)      b/((s+a)^2+(b^2))
e^(-at)cos(bt)      (s+a)/((s+a)^2+(b^2))



Properties
----------

    f(t)            F(s)
--------------------------------------
A f(t)              A F(s)
f1(t) + f2(t)       F1(s) + F2(s)
f1(t) - f2(t)       F1(s) - F2(s)
f(at)               1/a F(s/a) , a>0
f(t-t0) H[t-t0]     e^(-t0s) F(s) , t0 >= 0
f(t) H[t-t0]        e^(-t0s) L[ f(t+t0) ]
e^(-at) f(t)        F(s+a)
t f(t)              - d/ds F(s)

		    /x
1/t f(t)            | F(<lambda>) d<lambda>
                    /s

/t
| f1(<lambda>) f2(t-<lambda>) d<lambda>    F1(s) F2(s)
/0

f1(t) @ f2(t)       F1(s) F2(s) , @ represents convolution
f1(t) f2(t)         F1(s) @ F2(s)





L [ dn/dt^n (f(t)) ] = s^n F(s) - s^(n-1) f(0) - s^(n-2) f`(0) - s^(n-3) f``(0) - ... ... -s^0 f^`(n-1)(0)




As usual, no guarentees are made as to the accuracy or obsolescence of this information.  Use at your own risk.