The Laplace transform is an integral transform that converts a function of time into a function of complex frequency. It is defined as the integral of the function multiplied by e-st from 0 to infinity. The Laplace transform is used to solve differential equations by converting them to algebraic equations. Some key properties of the Laplace transform include linearity, shifting theorems, differentiation and integration formulas, and methods for periodic and anti-periodic functions.

1. Laplace transform is an
integral transform named after the great
French mathematician, Pierre Simon De
Laplace. It regularly used to transforms a
function of a real variable t to a function of a
complex variable s (complex frequency). The
Laplace transform changes one signal into
another according to some fixed set of rules
or equations. It is used to convert differential
equations into algebraic equations.

2. There are certain steps which need to be followed in order to do a
Laplace transform of a time function. In order to transform a given
function of time f(t) into its corresponding Laplace transform, we have
to follow the following steps:
First multiply f(t) by 𝑒−𝑠𝑡
, s being a complex number (s = a + i b).
Integrate this product w.r.t time with limits as zero and infinity. This
integration results in Laplace transformation of f(t), which is denoted
by F(s).
The Laplace transform L, of a function f(t) , t > 0 is defined by
0
∞
𝑓 𝑡 𝑒−𝑠𝑡
𝑑𝑡 , the resulting expression is a function s and denoted by
F(s). The Laplace transform of f(t) is denoted by L[f(t)]
𝑖. 𝑒 𝐿 𝑓 𝑡 = 0
∞
𝑓 𝑡 𝑒−𝑠𝑡
𝑑𝑡 = F(s)

3. The time function f(t) is obtained back from the Laplace
transform by a process called inverse Laplace
transformation and denoted by 𝐿−1
The inverse transform of F(s) = 𝐿−1
𝐹 𝑠 =
𝐿−1
𝐿 𝑓 𝑡 = 𝑓(𝑡)

4. The main properties of Laplace Transform as follows:
1. Linearity:
Let C1, C2 be constants. f(t), g(t) be the functions of time, t,
then
𝐿 𝐶1 𝑓 𝑡 + 𝐶2 𝑔 𝑡 = 𝐶1 𝐿 𝑓 𝑡 + 𝐶2 𝐿[𝑔 𝑡 ]

5. 2. First shifting Theorem: (s- shifting)
If 𝐿 𝑓 𝑡 = 𝐹 𝑠 then 𝐿 𝑒 𝑎𝑡 𝑓 𝑡 = 𝐹(𝑠 − 𝑎) and 𝐿 𝑒−𝑎𝑡 𝑓 𝑡 = 𝐹(𝑠 + 𝑎)
Also 𝐿−1
𝐹 𝑠 − 𝑎 = 𝑒 𝑎𝑡
𝑓 𝑡 and 𝐿−1
𝐹 𝑠 + 𝑎 = 𝑒−𝑎𝑡
𝑓 𝑡
3. Second shifting Theorem: (t- shifting, time shifting)
If 𝐿 𝑓 𝑡 = 𝐹 𝑠 then the Laplace Transform of f(t) after the delay of
time, T is equal to the product of Laplace Transform of f(t) and e-st
𝑖. 𝑒 𝐿 𝑓 𝑡 − 𝑇 𝑢 𝑡 − 𝑇 = 𝑒−𝑠𝑡
𝐹(𝑠) , where 𝑢(𝑡 − 𝑇) is the unit
step function.
Note: 𝑢 𝑛(𝑡) is the Heaviside step function which is given by
𝑢 𝑡 − 𝑛 =
0 𝑖𝑓 𝑡 < 𝑛
1𝑖𝑓 𝑡 ≥ 𝑛
In particular, 𝑢 𝑡 =
0, 𝑖𝑓 𝑡 < 0
1, 𝑖𝑓 𝑡 ≥ 0

6. 4. Change of scale property:
If 𝐿 𝑓 𝑡 = 𝐹 𝑠 then
Frequency scaling - 𝐿 𝑓 𝑎𝑡 =
1
𝑎
𝐹(
𝑠
𝑎
)
Time scaling - 𝐿 𝑓
𝑡
𝑎
= 𝑎𝐹(𝑎𝑠)
5. Differentiation:
If 𝐿 𝑓 𝑡 = 𝐹 𝑠 then
t- Differentiation:
𝐿
𝑑 𝑛
𝑑𝑡 𝑛 𝑓 𝑡 = 𝑠 𝑛
𝐿 𝑓 𝑡 − 𝑠 𝑛−1
𝑓 0 − 𝑠 𝑛−2
𝑓′
0 −. . −𝑓 𝑛−1
(0)
If n = 1,𝐿
𝑑
𝑑𝑡
𝑓 𝑡 = 𝑠𝐿 𝑓 𝑡 − 𝑓 0 = 𝑠𝐹 𝑠 − 𝑓(0)
If n = 2,𝐿
𝑑2
𝑑𝑡2 𝑓 𝑡 = 𝑠2
𝐿 𝑓 𝑡 − 𝑠𝑓 0 − 𝑓′
0 = 𝑠2
𝐹 𝑠 − 𝑠𝑓 0 − 𝑓′
0
s- Differentiation: (differentiation of transform)
𝐿 𝑡𝑓 𝑡 = −
𝑑
𝑑𝑠
𝐿 𝑓 𝑡 = −
𝑑
𝑑𝑠
{𝐹 𝑠 }
In general 𝐿 𝑡 𝑛
𝑓 𝑡 = −1 𝑛 𝑑 𝑛
𝑑𝑠 𝑛 𝐿 𝑓 𝑡 = −1 𝑛 𝑑 𝑛
𝑑𝑠 𝑛 {𝐹 𝑠 }

7. 6. Integration:
If 𝐿 𝑓 𝑡 = 𝐹 𝑠 then
t- Integration:
𝐿 0
𝑡
𝑓(𝑡)𝑑𝑡 =
1
𝑠
𝐿 𝑓 𝑡
s- Integration: (integration of transform)
𝐿
𝑓 𝑡
𝑡
= 0
∞
𝐿[𝑓 𝑡 ] 𝑑𝑠
7. Initial value Theorem:
If 𝐿 𝑓 𝑡 = 𝐹 𝑠 then
lim
𝑡 → 0
𝑓(𝑡) =
lim
𝑠 → ∞
𝑠𝐹(𝑠)
8. Final value Theorem:
If 𝐿 𝑓 𝑡 = 𝐹 𝑠 then
lim
𝑡 → ∞
𝑓(𝑡) =
lim
𝑠 → 0
𝑠𝐹(𝑠)
9. Convolution:
If 𝐿 𝑓 𝑡 = 𝐹 𝑠 & 𝐿 𝑔 𝑡 = 𝐺(𝑠) then
𝑓 ∗ 𝑔 𝑡 = 0
𝑡
𝑓 𝑡 𝑔 𝑡 − 𝑢 𝑑𝑢 = 0
𝑡
𝑔 𝑡 𝑓 𝑡 − 𝑢 𝑑𝑢 = 𝐿 𝑓 𝑡 𝐿[𝑔 𝑡 ] = 𝐹 𝑠 𝐺 𝑠

9. Procedure:
1. Take the Laplace transform on both side of the given
differential equation.
2. Use the initial conditions , which gives an algebraic equation.
3. Solve the algebraic equation and get the value of L(y) in terms
of s, which is F(s).(𝑖. 𝑒 𝐿 𝑦 = 𝐹(𝑠))
4. Find y by taking inverse Laplace transformation,𝑦 = 𝐿−1[𝐹 𝑠 ],
which is the required solution.

10. Periodic Function:
A function f(t) is said to be periodic with period T (> 0) 𝑖𝑓 𝑓(𝑡 + 𝑇) = 𝑓(𝑡)
Anti-Periodic function:
A function f(t) is said to be anti-periodic with period T, if 𝑓(𝑡 + 𝑇) = −𝑓(𝑡)
for all t
𝑖. 𝑒 𝑓 𝑡 + 2𝑇 = 𝑓 𝑡 + 𝑇 + 𝑇 = −𝑓 𝑡 + 𝑇 = − −𝑓 𝑡 = 𝑓(𝑡)
Example :
1.𝑆𝑖𝑛 2𝜋 + 𝑡 = 𝑠𝑖𝑛𝑡, hence 𝑠𝑖𝑛𝑡 is periodic function with period 2𝜋.
2.𝑆𝑖𝑛 𝜋 + 𝑡 = −𝑠𝑖𝑛𝑡, hence 𝑠𝑖𝑛𝑡 is anti periodic function with period
𝜋.

11. Laplace Transform of Periodic Functions
If f(t) is periodic with period T > 0, then 𝐿 𝑓 𝑡 =
1
1−𝑒−𝑠𝑇 0
𝑇
𝑒−𝑠𝑡 𝑓 𝑡 𝑑𝑡
Laplace Transform of Anti-Periodic Functions
If f(t) is anti-periodic with period T > 0, then 𝐿 𝑓 𝑡 =
1
1+𝑒−𝑠𝑇 0
𝑇
𝑒−𝑠𝑡 𝑓 𝑡 𝑑𝑡