1. Fourier transforms represent a function as a sum of sinusoidal functions using integral transforms. The Fourier transform of a function f(x) is defined as an integral transform using a kernel function, with examples including the Laplace, Fourier, Hankel, and Mellin transforms. 2. The Fourier integral theorem states that if a function f(x) is piecewise continuous and differentiable, its Fourier transform represents the function as an integral using sinusoidal functions. 3. The Fourier transform and its inverse are defined by integrals using the function and a complex exponential kernel. Properties of Fourier transforms include linearity, shifting, scaling, and relationships between a function and its derivative or integral transforms.

1. FOURIER TRANSFORMS

2. INTEGRAL TRANSFORMS
The integral transform 𝐹 𝑠 of a function 𝑓(𝑥) is defined as
𝐼 𝑓 𝑥 = 𝐹 𝑠 = 𝑎
𝑏
𝑓 𝑥 𝑘 𝑠, 𝑥 𝑑𝑥
Where 𝑘(𝑠, 𝑥) is called kernel of 𝑓(𝑥)
Examples:
1.Laplace Transform 𝑘(𝑠, 𝑥) = 𝑒−𝑠𝑥
𝐿 𝑓 𝑥 = 𝐹 𝑠 = 0
∞
𝑓 𝑥 𝑒−𝑠𝑥 𝑑𝑥
2.Fourier Transform 𝑘(𝑠, 𝑥) = 𝑒 𝑖𝑠𝑥
𝐹 𝑓 𝑥 = 𝐹 𝑠 =
1
2𝜋 −∞
∞
𝑓 𝑥 𝑒 𝑖𝑠𝑥 𝑑𝑥
3.Hankel Transform 𝑘(𝑠, 𝑥) = 𝑥𝐽 𝑛(𝑠, 𝑥)
𝐻 𝑓 𝑥 = 𝐹 𝑠 = 0
∞
𝑓 𝑥 𝑥𝐽 𝑛(𝑥) 𝑑𝑥
4.Millin Transform 𝑘 𝑠, 𝑥 = 𝑥 𝑠−1
𝑀 𝑓 𝑥 = 𝐹 𝑠 = 0
∞
𝑓 𝑥 𝑥 𝑠−1 𝑑𝑥

3. FOURIER INTEGRAL
THEOREM
If 𝑓(𝑥) is piece-wise continuously, differentiable and
absolutely integrable in (−∞, ∞)
Then 𝑓 𝑥 =
1
2𝜋 −∞
∞
−∞
∞
𝑓 𝑡 𝑒 𝑖𝑠 𝑥−𝑡 𝑑𝑡𝑑𝑠
Or 𝑓 𝑥 =
1
𝜋 0
∞
−∞
∞
𝑓 𝑡 cos 𝑠 𝑡 − 𝑥 𝑑𝑡𝑑𝑠
This is called Fourier integral theorem/Formula.

4. FOURIER TRANSFORM
If 𝑓(𝑥) is defined in(−∞, ∞), then the
Fourier Transform of 𝑓(𝑥) is given by
𝐹 𝑓 𝑥 = 𝐹 𝑠 =
1
2𝜋 −∞
∞
𝑓 𝑥 𝑒 𝑖𝑠𝑥
𝑑𝑥

5. INVERSE FOURIER
TRANSFORM
The inverse Fourier Transform of
𝐹 𝑠 is given by
𝑓 𝑥 =
1
2𝜋 −∞
∞
𝐹 𝑠 𝑒−𝑖𝑠𝑥
𝑑𝑠

6. Fourier Sine Transform
The Fourier Sine Transform is given by
𝐹𝑠 𝑓 𝑥 = 𝐹𝑠 𝑠 =
2
𝜋 0
∞
𝑓 𝑥 𝑠𝑖𝑛𝑠𝑥 𝑑𝑥
Inverse Fourier Sine Transform
The Inverse Fourier Sine Transform of 𝐹𝑠 𝑠 is given by
𝑓 𝑥 =
2
𝜋 0
∞
𝐹𝑠(𝑠)𝑠𝑖𝑛𝑠𝑥 𝑑𝑠
Fourier Cosine Transform
The Fourier Cosine Transform is given by
𝐹𝑐 𝑓 𝑥 = 𝐹𝑐 𝑠 =
2
𝜋 0
∞
𝑓 𝑥 𝑐𝑜𝑠𝑠𝑥 𝑑𝑥
Inverse Fourier Cosine Transform
The Inverse Fourier Cosine Transform of 𝐹𝑐 𝑠 is given
by
𝑓 𝑥 =
2
𝜋 0
∞
𝐹𝑐(𝑠)𝑐𝑜𝑠𝑠𝑥 𝑑𝑠

7. PROPERTIES OF FOURIER
TRANSFORMS1.Linearity Property
𝐼𝑓 𝐹 𝑓 𝑥 = 𝐹 𝑠 𝑎𝑛𝑑 𝐹 𝑔 𝑥 = 𝐺 𝑠 , 𝑡ℎ𝑒𝑛 𝐹[𝑎𝑓(𝑥) + 𝑏𝑔(𝑥)] = 𝑎𝐹(𝑠) + 𝑏𝐺(𝑠)
2.Shifting Property
𝐼𝑓 𝐹 𝑓 𝑥 = 𝐹 𝑠 , 𝑡ℎ𝑒𝑛 𝐹[𝑓(𝑥 − 𝑎)] = 𝑒 𝑖𝑎𝑥
𝐹(𝑠)
3.If 𝐹 𝑠 =
1
2𝜋 −∞
∞
𝑓 𝑥 𝑒 𝑖𝑠𝑥
𝑑𝑥, then 𝐹 𝑒 𝑖𝑎𝑥
𝑓 𝑥 = 𝐹[𝑎 + 𝑠]
4.Change of scale Property
𝐼𝑓 𝐹 𝑓 𝑥 = 𝐹 𝑠 , 𝑡ℎ𝑒𝑛 𝐹 𝑓 𝑎𝑥 =
1
𝑎
𝐹(
𝑠
𝑎
), 𝑎 ≠ 0
5.𝐼𝑓 𝐹 𝑓 𝑥 = 𝐹 𝑠 , 𝑡ℎ𝑒𝑛 𝐹 𝑥 𝑛
𝑓 𝑥 = −1 𝑛 𝑑 𝑛
𝑑𝑠 𝑛 ( 𝐹 𝑠 )
6.𝐼𝑓 𝐹 𝑓 𝑥 = 𝐹 𝑠 𝑎𝑛𝑑 𝑓 𝑥 ⟶ 0 𝑎𝑠 𝑥 ⟶ ±∞, 𝑡ℎ𝑒𝑛 𝐹 𝑓′
𝑥 = −𝑖𝑠𝐹(𝑠)
7.𝐼𝑓 𝐹 𝑓 𝑥 = 𝐹 𝑠 , 𝑡ℎ𝑒𝑛 𝐹[ 𝑎
𝑥
𝑓 𝑥 𝑑𝑥] =
𝐹 𝑥
−𝑖𝑠
8.𝐼𝑓 𝐹 𝑓 𝑥 = 𝑓 𝑠 , 𝑡ℎ𝑒𝑛 𝐹 𝑓 −𝑥 = 𝐹(−𝑠)
9.𝐼𝑓 𝐹 𝑓 𝑥 = 𝐹 𝑠 , 𝑡ℎ𝑒𝑛 𝐹 𝑓 𝑥 = 𝐹(−𝑠)
10.𝐼𝑓 𝐹 𝑓 𝑥 = 𝐹 𝑠 , 𝑡ℎ𝑒𝑛 𝐹 𝑓 −𝑥 = 𝐹(𝑠)
11.Modulation Property
𝐼𝑓 𝐹 𝑓 𝑥 = 𝐹 𝑠 , 𝑡ℎ𝑒𝑛 𝐹 𝑓 𝑥 𝑐𝑜𝑠𝑎𝑥 =
1
2
[𝐹 𝑠 − 𝑎 + 𝐹 𝑠 + 𝑎 ]

8. CONVOLUTION OF TWO
FUNCTIONS
The convolution of two functions 𝑓(𝑥)
and 𝑔(𝑥)
is defined by
𝑓 ∗ 𝑔 =
1
2𝜋 −∞
∞
𝑓 𝑡 𝑔 𝑡 − 𝑥 𝑑𝑡

9. Convolution Theorem
𝐹(𝑓 ∗ 𝑔) = 𝐹(𝑠). 𝐺(𝑠) where 𝑓 and 𝑔 are two
functions and 𝐹(𝑠) =
1
2𝜋 −∞
∞
𝑓 𝑥 𝑒 𝑖𝑠𝑥
𝑑𝑥
& 𝐺(𝑠) =
1
2𝜋 −∞
∞
𝑔 𝑥 𝑒 𝑖𝑠𝑥
𝑑𝑥
Parseval’s Identity
A function f(x) and its transform F(s) satisfy the
identity
−∞
∞
𝑓 𝑥 2
𝑑𝑥 = −∞
∞
𝐹 𝑠 2
𝑑𝑠

10. PROPERTIES OF FOURIER SINE AND
COSINE TRANSFORMS
1.(i). 𝐹𝑠 𝑎𝑓 𝑥 + 𝑏𝑔 𝑥 = 𝑎𝐹𝑠 𝑓 𝑥 + 𝑏𝐹𝑠[𝑔 𝑥 ]
(ii) 𝐹𝑐 𝑎𝑓 𝑥 + 𝑏𝑔 𝑥 = 𝑎𝐹𝑐 𝑓 𝑥 + 𝑏𝐹𝑐[𝑔 𝑥 ]
2.(i). 𝐹𝑠 𝑓 𝑥 𝑐𝑜𝑠𝑎𝑥 =
1
2
[𝐹𝑠 𝑠 + 𝑎 + 𝐹𝑠 (𝑠 − 𝑎)]
(ii) 𝐹𝑐 𝑓 𝑥 𝑐𝑜𝑠𝑎𝑥 =
1
2
[𝐹𝑐 𝑠 + 𝑎 + 𝐹𝑐 𝑠 − 𝑎 ]
3.(i). 𝐹𝑠 𝑓 𝑥 𝑠𝑖𝑛𝑎𝑥 =
1
2
[𝐹𝑐 𝑠 − 𝑎 − 𝐹𝑐 (𝑠 + 𝑎)]
(ii) 𝐹𝑐 𝑓 𝑥 𝑠𝑖𝑛𝑎𝑥 =
1
2
[𝐹𝑠 𝑠 + 𝑎 − 𝐹𝑠 𝑠 − 𝑎 ]
4.(i). 𝐹𝑠 𝑓 𝑎𝑥 =
1
𝑎
𝐹𝑠
𝑠
𝑎
(ii) 𝐹𝑐 𝑓 𝑎𝑥 =
1
𝑎
𝐹𝑐
𝑠
𝑎
5.(i). 𝐹𝑠 𝑓′ 𝑥 = −𝑠𝐹𝑐 𝑠 , 𝑖𝑓 𝑓 𝑥 ⟶ 0𝑎𝑠 𝑥 ⟶ ∞
(ii) 𝐹𝑐 𝑓′ 𝑥 =
−
2
𝜋
𝑓 0 + 𝑠𝐹𝑠 𝑠 , 𝑖𝑓 𝑓 𝑥 ⟶ 0𝑎𝑠 𝑥 ⟶ ∞
6.(i). 𝐹𝑠 𝑓′′ 𝑥 =
2
𝜋
𝑠𝑓 0 − 𝑠2
𝐹𝑠 𝑠 , 𝑖𝑓 𝑓 𝑥 &𝑓′(𝑥) ⟶ 0𝑎𝑠 𝑥 ⟶ ∞
(ii) 𝐹𝑐 𝑓′′ 𝑥 =
−
2
𝜋
𝑓′ 0
− 𝑠2
𝐹𝑐 𝑠 , 𝑖𝑓 𝑓 𝑥 &𝑓′(𝑥) ⟶ 0𝑎𝑠 𝑥 ⟶ ∞
7.(i). 𝐹𝑠 𝑥𝑓 𝑥 = −
𝑑
𝑑𝑠
(𝐹𝑐 𝑓 𝑥 )
(ii) 𝐹𝑐 𝑥𝑓 𝑥 = −
𝑑
𝑑𝑠
(𝐹𝑠 𝑓 𝑥 )

11. IDENTITIES
𝟎
∞
𝑭 𝒔 𝒔 𝑮 𝒔 𝒔 𝒅𝒔 = 𝟎
∞
𝒇 𝒙 𝒈 𝒙 𝒅𝒙
𝟎
∞
𝑭 𝒄 𝒔 𝑮 𝒄 𝒔 𝒅𝒔 = 𝟎
∞
𝒇 𝒙 𝒈 𝒙 𝒅𝒙
Parseval’s Identity
𝟎
∞
𝑭 𝒔 𝒔 𝟐 𝒅𝒔 = 𝟎
∞
𝑭 𝒄 𝒔 𝟐 𝒅𝒔 = 𝟎
∞
𝒇 𝒙 𝟐 𝒅𝒙