These are the updated and easily understandable applications of Bessel's function. Including Daily life Applications.

1. Applications of Bessel’s
Function.

2. Contents.
>History and definition.
>Introduction.
>Application in Mechanical Engineering.
>Uses of Bessel’s function in frequency
modulation.
>Application of Bessel’s Function is
Biology.
>Kettle Drum.
>Application of Bessel’s function in daily
life.

3. Introduction of Bessel’s
Function.
Presenting By Shazia Shahzad.

4. Bessel’s Function:-
History:-
Bessel function was first defined by the mathematician Daniel
Bernoulli and the generalized by Friedrich Bessel in 18th century.
Definition:-
Bessel equation are the solution y(x) of differential equation
𝑥2 𝑑2𝑦
𝑑𝑥2 + 𝑥
𝑑𝑦
𝑑𝑥
+ 𝑥2
− α2
𝑦 = 0

5. Introduction:.

6. APPLICATION OF BESSEL’S
FUNCTION IN
MECHANICAL
ENGINEERING
Presenting by Sarah Ashraf.

7. HEAT TRANSFER IN A
CIRCULAR FIN:

8. HEAT TRANSFER IN A
CIRCULAR FIN

9. USE OF BESSEL’S
FUNCTION IN
FREQUENCY
MODULATION

10. FREQUENCY MODULATION AND BESSEL
FUNCTION:
𝑡 ))
𝑡 )) )𝑡)

11. COMPONENT AMPLITUDES:

12. COMPONENT
AMPLITUDES:

13. COMPONENT AMPLITUDES:

15. “cos(𝟐𝝅𝒇𝒄𝒕 + 𝜷sin(𝟐𝝅𝒇𝒎𝒕 )) =
−∞
∞
𝑱𝑲 (𝜷) cos(𝟐𝝅(𝒇𝒄 +𝒌 𝒇𝒎)𝒕)”
𝑡 ))

16. cos(𝟐𝝅𝒕 + 𝜷(𝟐𝝅𝒇𝒎𝒕 )) – sin(𝟐𝝅𝒇𝒄𝒕 )sin(𝜷sin(𝟐𝝅𝒇𝒎𝒕 ))
cos(𝟐𝝅𝒇𝒄𝒕 cos( 𝜷sin(𝟐𝝅𝒇𝒎𝒕 ))
𝒋𝟎(𝜷)cos(𝟐𝝅𝒇𝒄𝒕) +
𝒌=𝟏
∞
𝒋𝟐𝒌(𝜷) {cos(𝟐𝝅(𝒇𝒄 − 𝟐𝐤𝒇𝒎)𝒕) + cos(𝟐𝝅(𝒇𝒄 − 𝟐𝒌𝒇𝒎)𝒕}
𝒏(𝒆𝒗𝒆𝒏)
𝒋𝒏 (𝜷)cos(𝟐𝝅 (𝒇𝒄 + 𝐧𝒇𝒎)𝒕)
sin(𝟐𝝅𝒇𝒄𝒕 )sin(𝜷sin(𝟐𝝅𝒇𝒎)𝒕 )
𝒕𝒐
−
𝒏(𝒐𝒅𝒅)
𝒋𝒏 (𝜷)cos(𝟐𝝅 (𝒇𝒄 + 𝐧𝒇𝒎)𝒕)
𝑪𝒐𝒔(𝟐𝝅𝒇𝒄𝒕 + 𝜷sin(𝟐𝝅𝒇𝒎𝒕 )) = =
−∞
∞
𝑱𝑲 (𝜷)cos(𝟐𝝅(𝒇𝒄 +𝒌 𝒇𝒎).

17. Applications of Bessel’s
Function in BIOLOGY
Presenting By Hafiza
Maheen Ajmal.

18. Bessel function in the structure of DNA
Bessel's function is given by:
𝐽𝛼 = 𝐽𝛼(𝑥) =
(−1)𝑛
𝑛!
(𝑛 + 𝛼)! × (
𝑥
2
)(2𝑛 + 𝛼)𝐽𝛼 = 𝐽𝛼(𝑥)
=
(−1)𝑛
𝑛!
(𝑛 + 𝛼)! × (
𝑥
2
)(2𝑛 + 𝛼)
Where, variable α is called the order of the function, and the
values of n are integers.
So, now how does one plot the Bessel function? Simple, if you
want to plot the Bessel function of order zero, you plug in the
values 𝛼 = 0 and 𝑛 = 0 and then plot J as a function of 𝑥 over
some range −𝑥 to +𝑥. Next you plug in 𝛼 = 0 and 𝑛 = 1, plot
again, and add the resulting curve to the one for which n = 0, just
as curves were added together to give the Fourier sum:

19. Thus, as the order increases, the position of
the first peak of the function occurs farther
from the origin.
Francis Crick showed in his doctoral
dissertation that in the transform of a
continuous helix, the intensity along a layer
line is described by the square of the Bessel
function whose order α equals the number l
of the layer line..
Thus, the intensity of the central line,
layer line zero, varies according
to J0(x)]2J0(x)]2, which is the square
of the above equation, α = 0 (red). The
intensity of the first line above (or
below) center varies according
to J1(x)]2J1(x)]2(green), and so on..

20. Kettle Drum
Presenting By Kinza Nadeem

21. Kettle drum:

22. Kettle Drum.

24. Kettle Drum

25. Applications of
Bessel’s
Function in
daily life.
Presenting By Kubra Ghaffar.

26. Continue..

27. Continue..
The OTF for out-of-focus blur is
given by;
𝐻 𝑖, 𝑗 =
𝐽1𝑅𝑝
𝑅𝑝
, 𝑝2 = 𝑖2 + 𝑗2.
𝐻 𝑖, 𝑗 =
𝑒𝑅𝑝
2𝜋𝑅𝑝

28. Continue…

29. THANK U for your attention.
Now you can clap
If You have any Questions.
Just Google them.