The document defines the Laplace transform as an integral that transforms a function defined for all positive real numbers into another function of a complex variable. It states that the inverse Laplace transform is a complex integral that transforms the function back. It provides examples of the linearity property of the Laplace transform and uses it to prove the first shifting theorem. The linearity property is then applied to decompose functions using Euler's formula.

1. By
Prof. K.K.Pokar
GEC BHUJ
LAPLACE TRANSFORM

2. LAPLACE TRANSFORM:
DEFINITION
Let f(t) be given function defined for all t 0,
Then it’s Laplace Transform is defined as the following integral if
exists
Transform Given
function
Laplace Transform is an improper

3. INVERSE LAPLACE TRANSFORM:
If then
Inverse TransformGiven
function
Inverse Laplace Transform is a complex integral

4. BASIC EXAMPLES

5. The Laplace Transform is a Linear operation
i.e.,
LINEARITY OF LAPLACE
TRANSFORM:
= RHS

6. • Statement :
If then
OR
Inverse form:
PROOF:
FIRST SHIFTING THEOREM OR
S-SHIFTING THEOREM OF LAPLACE
TRANSFORM

7. APPLICATIONS OF LINEARITY
By Linearity of Laplace
Transform

8. APPLICATIONS OF LINEARITY
By Euler’s Formula
and
By comparing real and imaginary parts of both sides we get