Annotated and linked table of Laplace Transforms


y1(t)+y2(t)
Cy(t) (C is a constant)
Y1(s)+Y2(s)
CY(s)
Linearity
This is used everywhere. First official mention
tn
n is a non-negative integer
n!/sn+1 Done for n=0; done for n=1; done for n=2; statement for any n. Used very often afterwards.
eat 1/(s-a) Verification. Used often.
sin(kt)
cos(kt)
k/(s2+k2)
s/(s2+k2)
Verification for cosine using complex exponents. The text proves this using integration by parts. Both results are used often. You need to know and believe in Euler's formula.
sinh(kt)
cosh(kt)
k/(s2-k2)
s/(s2-k2)
Not verified in class, but a discussion here indicates why sinh and cosh are useful.

y(t)=t if t in [0,1] else y(t)=0.
(-se-s-e-s+1)/s2 Computed from the definition. Students should also know how to do this with U and the translation theorems.
y(t) is a function
with exponential growth
Y(s)-->0 as s-->infinity.
Y(s)-->the net area under all of the y(t) function as s-->0+.
Asymptotics
Discussed here. Please note this does not apply to the Dirac delta function, which, in spite of its name, is not a function.
y´(t) sY(s)-y(0) Verified here. Used later frequently.
Solving 1y´(t)+2y(t)=3cos(t)
with y(0)=4
-4+sY(s)+2Y(s)=3s/(s2+1) First use of Laplace transform method to solve an initial value problem. The steps (transform, solve, inverse transform) are used very often, so this should be considered a model. Here is a second order equation, and here is a link to a solution of an integral equation from the course I gave a year ago, since the choice I made this year didn't work out.
y(n)(t) snY-sn-1y(0)-sn-2y´´(0)-...-y(n-1)(0) First mentioned here and first used here to solve a second order ODE.
eatf(t) F(s-a) First translation theorem. Examples. Many other examples later, such as here and here.
U(t)
U(t)=0 if t<0 & =1 if t>=0.
1/s Defined here and its use in writing piecewise functions algebraically follows immediately afterwards.
f(t-a)U(t-a) e-asF(s) Second translation theorem. Examples.
g(t)U(t-a) e-as multiplied by the Laplace transform of g(t+a). Alternate form of the second translation theorem Examples. Many other examples later, such as here and here.
t·f(t) F´(s) Verification. Used here.
(-1)ntn·f(t) (dn/dsn)F(s) Just the statement
f*g(t)=0tf(tau)g(t-tau)dtau F(s)·G(s) Convolution defined. One and two direct computations of convolution. and here's the statement of the Laplace transform fact. And some simple uses. A more complicated use is attempted here.
0tf(tau)dtau F(s)/s Verified here
delta(t-t0) e-t0s Verification here. Before and after this are substantial discussions of the "theory", and then some examples using delta in mathematical models.


Maintained by greenfie@math.rutgers.edu and last modified 9/20/2005.