The document discusses the Laplace transform, an integral transform developed by Pierre-Simon Laplace and later advanced by Oliver Heaviside, which converts functions of real variables into complex variables. It outlines the types of Laplace transforms, their applications in science and engineering, particularly in solving differential equations, and mentions both advantages and disadvantages of using Laplace transforms. The importance of these transforms in electrical engineering is emphasized, noting their effectiveness in analyzing linear time-invariant systems.

1. ﷽

2. National College of Business
Administration & Economics

3. Group Members
• Muhammad Ahsan
• M Shahid Zafar
• Wamiq Ali
• M Sarmad Shuja
• Zain Abid

4. Laplace
Transform
Multi Variable
Calculus

5. History
• Invented
• Discovered
• Laplace mean

6. Invented
• Laplace transform, a particular integral transform invented by the French mathematician
Pierre-Simon Laplace.
• Systematically developed by the British physicist Oliver Heaviside.

7. Discovered
• Pierre-Simon Laplace was a prominent French mathematical physicist and
astronomer of the 19th century, who made crucial contributions in the
arena of planetary motion by applying Sir Isaac Newton's theory of
gravitation to the entire solar system.

8. Laplace Mean
• The Laplace transform is an integral transform named after its inventor
Pierre-Simon Laplace (/ləˈplɑːs/). It transforms a function of a real variable
t (often time) to a function of a complex variable s (complex frequency).

9. Definition
• The conversion in mathematics, It transforms a function of a real
number t (often time) to a function of a complex variable
s(complex frequency).
• In other word the conversion of time to complex frequency is
called Laplace transform. The Laplace has many applications in
science and engineering.

10. Types
• Unilateral Transform
• Bilateral Transform
• Inverse Transform

11. Unilateral Transform
• The unilateral Laplace transform of any signal is
identical to its bilateral Laplace transform.
𝑓 6 =
0
∞
𝑓 𝑡 𝑒−𝑠𝑡 𝑑𝑡

12. Bilateral transform
• The two-sided Laplace transform or bilateral Laplace transform is an
integral transform equivalent to probability's moment generating function.
• Two-sided Laplace transforms are closely related to the Fourier transform.
𝑓 6 =
−∞
∞
𝑓 𝑡 𝑒−𝑠𝑡 𝑑𝑡

13. Inverse transform
• A Laplace transform which is a constant multiplied by a function has an
inverse of the constant multiplied by the inverse of the function.
• First shift theorem: L − 1 { F ( s − a ) } = e a t f ( t ) , where f(t) is the inverse
transform of F(s).

14. Examples
1) Find Laplace transform of f(t),
Where,
f(t) = 3, 0 < t <5
f(t) = 0, t > 5
By definition Laplace transform

15. Examples
2) F(t) = t, for 0 < t < 4 and f(t) = 5 for t > 4
By definition of Laplace transform

16. Examples
3) Find Laplace transform of f(t), where f(t) = K L(K), we must find.
By definition
= 𝐋{𝐟 𝒕 } =
𝟎
∞
𝒆−𝒔𝒕 𝒇(𝒕)𝒅𝒕

17. Uses
• Why does we use?
• Where are Laplace transforms used?

18. Why does we use?
• The Laplace Transform is a
generalized Fourier Transform, since
it allows one to obtain transforms of
functions that have no Fourier
Transforms.

19. Where are Laplace transforms used?
The Laplace transform can also be used to solve differential equations and is
used extensively in electrical engineering.
1) Basically, a Laplace transform will convert a function in some domain
into a function in another domain without the changing the values of the
function.
2) Since equations having polynomials are easier to solve, we employ
Laplace transform to make calculation easier.

20. Importance of Laplace Transforms
• In electrical circuits, a Laplace transform is
used for the analysis of linear time-
invariant systems.
• Laplace transform is widely used by
Electronics engineers to quickly solve
differential equations occurring in the
analysis of electronic circuits.

21. Advantages
• Laplace transforms methods offer the following
advantages over the classical methods.
• Initial conditions are automatically considered in the
transformed equations.
• Much less time is involved in solving differential
equations.
• It gives systematic and routine solutions for differential
equations.

22. Disadvantages
• Unsuitability for data processing in random
vibrations.
• Analysis of discontinuous inputs.
• Inability to exist for few Probability Distribution
Functions

23. Thank You 