### 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.