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.

e^at 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-s^n-1y(0)-s^n-2y´´(0)-...-y^(n-1)(0) First mentioned here and first used here to solve a second order ODE.

e^atf(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)=₀^tf(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.

₀^tf(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.

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.
e^at	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-s^n-1y(0)-s^n-2y´´(0)-...-y^(n-1)(0)	First mentioned here and first used here to solve a second order ODE.
e^atf(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)=₀^tf(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.
₀^tf(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.