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![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..](https://image.slidesharecdn.com/m-210404182949/75/Applications-of-Bessel-s-Function-19-2048.jpg)
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- 2. Contents. >History and definition. >Introduction. >Applicationin 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. PresentingBy Shazia Shahzad.
- 4. Bessel’s Function:- History:- Bessel functionwas 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.
- 6. APPLICATION OF BESSEL’S FUNCTIONIN MECHANICAL ENGINEERING Presenting by Sarah Ashraf.
- 7. HEAT TRANSFER INA CIRCULAR FIN:
- 8. HEAT TRANSFER INA CIRCULAR FIN
- 9. USE OF BESSEL’S FUNCTIONIN FREQUENCY MODULATION
- 10. FREQUENCY MODULATION ANDBESSEL FUNCTION: 𝑡 )) 𝑡 )) )𝑡)
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- 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 Functionin BIOLOGY Presenting By Hafiza Maheen Ajmal.
- 18. Bessel function inthe 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 theorder 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 ByKinza Nadeem
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- 25. Applications of Bessel’s Function in dailylife. Presenting By Kubra Ghaffar.
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- 27. Continue.. The OTF forout-of-focus blur is given by; 𝐻 𝑖, 𝑗 = 𝐽1𝑅𝑝 𝑅𝑝 , 𝑝2 = 𝑖2 + 𝑗2. 𝐻 𝑖, 𝑗 = 𝑒𝑅𝑝 2𝜋𝑅𝑝
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- 29. THANK U foryour attention. Now you can clap If You have any Questions. Just Google them.