The fresnel integrals solved using contour integration

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Old school complex integration techniques employed to solve the fresnel integrals.

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Hi,

"ok, cool" : 2:42, 4:15, 6:26, 7:52, 11:27, 12:18,

"terribly sorry about that" : 8:53, 9:16, 11:09, 12:52 .

CM_France

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!

HeyKevinYT

This was how I first learnt how to solve them! 😊

edmundwoolliams

This video made contour integrals click for me!

bennettkinder

What app are you using? It looks super smooth.

whatRedditSaysToday

12:18 Doesn't that require the use of DCT or MCT? To be able to switch the order of the limit operation and integration? The function doesn't seem to be monotone, taking MCT out of the picture, so how would DCT work here? Or is there perhaps a different theorem that comes into play?

الْمَذْهَبُالْحَنْبَلِيُّ-تذ

How do you know when to sue a specific contour because some have semi-circle or box or pizzaslice or keyhole, etc. and I don't understand when to use which

redroach