The fresnel integrals solved using contour integration
19K views · Apr 14, 2024 · Education
Comments · 53
@HeyKevinYT · 2 years ago (edited)
I learned these integrals last week in my complex analysis course! My professor did the general case of the integrals from 0 to ∞ of cos ax² dx (call it I₁) and of sin ax² dx (call it I₂), with a > 0. This means I₁ + iI₂ = integral from 0 to ∞ of exp(iax²) dx (call it I₃). We then set up the same contour and path parameterizations that you did and obtained the expression for I₃, whose real part and imaginary part are equal = I₁ = I₂ = √2/4 * √(π/a), and the case of a = 1 gives the solution as in the video. A slightly different approach for the definition of f and still very elegant!
18
@CM63_France · 2 years ago
Hi,<br><br>"ok, cool" : <a href="https://www.youtube.com/watch?v=PvdVCnDbmtU&t=162">2:42</a> , <a href="https://www.youtube.com/watch?v=PvdVCnDbmtU&t=255">4:15</a> , <a href="https://www.youtube.com/watch?v=PvdVCnDbmtU&t=386">6:26</a> , <a href="https://www.youtube.com/watch?v=PvdVCnDbmtU&t=472">7:52</a> , <a href="https://www.youtube.com/watch?v=PvdVCnDbmtU&t=687">11:27</a> , <a href="https://www.youtube.com/watch?v=PvdVCnDbmtU&t=738">12:18</a> ,<br><br>"terribly sorry about that" : <a href="https://www.youtube.com/watch?v=PvdVCnDbmtU&t=533">8:53</a> , <a href="https://www.youtube.com/watch?v=PvdVCnDbmtU&t=556">9:16</a> , <a href="https://www.youtube.com/watch?v=PvdVCnDbmtU&t=669">11:09</a> , <a href="https://www.youtube.com/watch?v=PvdVCnDbmtU&t=772">12:52</a> .
31
@edmundwoolliams1240 · 2 years ago
This was how I first learnt how to solve them! 😊
7
@bennettkinder5654 · 2 years ago
This video made contour integrals click for me!
3
@Ramana-k5m · 1 year ago
Thank you Kamaal Mirza for The two fresnel integrals solving by using contour integration.
@jojolov1234 · 2 years ago
Beautiful beyond words 😍
2
@HamzaAl-Kukhun · 7 months ago (edited)
Fantastic!
@sIndus5942 · 1 year ago
Clear conception. ❤ U from India🇮🇳🪷
@RocketsNRovers · 2 years ago
i loved it
@Impedancerupnath1234 · 2 years ago (edited)
wonderful teaching sir
8
@samr3100 · 1 year ago
What allows you to move the limit inside the integral for the capital gamma part?
1
@z3bka · 2 years ago
could you do it using Imaginary and Real parts of (e^i(x^2))
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