Laplace And Inverse Laplace Transforms

Let a function f be defined for t≥0. Then, Laplace Transform of f is defined as:

for those values of s where this integral exists.

Also, the inverse Laplace Transform is defined as:

First Shifting Theorem: If L(f(t))=F(s), then L(eatf(t))=F(s−a).

Laplace Transforms Of Elementary Functions

The unit step function ua(t) is defined as:

Laplace Transform of ua(t),

Second Shifting Theorem: If L(f(t))=F(s), then

Theorem: Let f(t) be continuous for t≥0 and is of exponential order. Further, suppose that f is differentiable with f′ piecewise continuous in [0,∞). Then,

Also,

L(f(n))=snL(f)-sn−1f(0)-sn−2f(1)(0)-⋯-f(n−1)(0)

Theorem: Let F(s) be the Laplace Transform of f. If f is piecewise continuous in [0,∞) and is of exponential order, then

Theorem: Let F(s) be the Laplace Transform of f, then

Theorem: Let F(s) be the Laplace Transform of f and the limit of f(t)/t exists as t→0+, then

Initial Value Problems

Theorem for Periodic Functions: Let f:[0,∞)→R be a periodic function with period T>0, i.e., f(t+T)=f(t) ∀ t≥0. Then,

Convolution: Let f and g be two functions defined in [0,∞).

Then, the convolution of f and g,is defined by:

Convolution Theorem: The convolution f∗g has the Laplce Transform property:

Laplace And Inverse Laplace Transforms

Initial Value Problems