Fourier Transform,and Integral Transform

1. • History of Laplace Transform
• History of Fourier Transform
• History of Hankel Transform
• Hankel Transform

2. In mathematics, an integral transform maps a function
from its original function space into another function
space via integration, where some of the
properties of the original function might be more easily
characterized and manipulated than in the original
function space.

3.  The Laplace transform has many applications in
science and engineering because it is a tool for solving
Differential equations.
 It is an integral transform that converts a function of a
real variable to a function of complex variable.
 This was 1st introduce by a French Mathematician
Laplace in the year 1790 in his work on probability
theorem.
 This technique became popular when Heaviside
applied to the solution of ODEs representing a
problem in Electrical Engineering.

4.  Analysis of Electronic Circuit
 System Modeling
 Signal Processing

5.  Fourier Transform is invented by a French
Mathematician in 1807 to explain the flow of heat
around anchor ring. He showed that any periodic
signal could be represented by a series of sinusoidal
function.
 It is a powerful tool in diverse field of science and
engineering .
 FT has become indispensable in the numerical
calculations needed to design electrical circuits, to
analyze mechanical vibrations and to study wave
propagation.

6.  Signal Processing
 Digital Image Processing
 Cryptography
 Vibration Analysis
 Oceanography
 Sonar

7.  The Hankel Transform is an integral transform and
was developed by the mathematician Herman Hankel
which occurs in the study of function which depend
only on the distance from the origin. It is also known
as the Fourier-Bessel Transform.

8.  Hankel transform expresses any given function f(r)
as the weighted sum of an infinite number
of Bessel functions of the first kind Jν(kr).
 The Fourier transform for an infinite interval is
related to the Fourier series over a finite interval,
so the Hankel transform over an infinite interval is
related to the Fourier–Bessel series over a finite
interval.

9. Bessel Function
The Bessel function are a series of solutions to a
second order differential equation 𝑥2
𝑦′′
+ 𝑥𝑦′
+
𝑥2 + 𝑝2 𝑦 = 0 that arises in many diverse situation.
Fourier–Bessel series
Fourier–Bessel series is a particular kind
of generalized Fourier series (an infinite
series expansion on a finite interval) based on Bessel
functions.

10.  The Hankel transform appears when one write the
multidimensional Fourier transform in hypo spherical
coordinate, which is the reason why the Hankel
transform often appears in physical problems with
cylindrical or spherical symmetry.

11. The use of Hankel transform has many advantages.
 It is applicable to both homogeneous and
inhomogeneous problems.
 It simplify calculations and singles out the purely
computational part of solution.s