Annotated and linked table of Laplace Transforms

Cy(t) (C is a constant)
This is used everywhere. First official mention
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.
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.
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+.
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)=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 and last modified 9/20/2005.