Complex Analysis: Integral of sin(x)/x using Contour Integration

Today, we use complex analysis to evaluate the improper integral of sin(x)/x, also known as the Dirichlet Integral.

Laplace Transform Method:

[qncubed]

For those who are unsatisfied with the explanation at around 15:20, here is an alternative way to evaluate the integral over Gamma, which requires further estimations.

[metincanatas]

Every theorem is proved while solving question, very nice.

[juniorcyans]

Do you have to rewrite the interval from negative infinity to positive infinity as 2* from zero to infinity? I don’t see the point.

[icee]

noice All this just to prove inverse fourier transform of Fourier transform is the function itself .

[flix]

Thanks a ton! you're a blessing for high schoolers like me who want to learn contour integration

[alicesmith]

This is so cool! Thank you so much for this example; I was quite lost when my professor brought this kind of process up as a point tangential to the lecture.

[edwardperry]

Great video. Someday I'm going to teach a class in how to properly write the letter "x." Let me know if you'd like to attend.

[abcdef]

lets say epsilon be r to reduce words
at 10:35 integ [ exp(i r exp (i * t)) ] ~ 1/r (1 - exp(- r * constant)) when r goes to 0, this results in 0/0 and, this was the same as the jordan thing. or he could have done the same just to apply R going to infinity BEFORE the integration, but he didnt. but ok the limit goes to ~ r const / 1 = 0

[peterastor]

Loved your series (no pun intended) of math videos. In your videos on the integral of sin(x)/x, I wondered whether there was a simple answer if the limits of integration went from a (>0) to infinity, rather than from 0 to infinity.

[maalikserebryakov]

What im most interested in is

Issue 1 - How do you convert from a real function to a corresponding complex function?
Here you changed sin(x) to e^iz
and I do not see why.
And you dropped the “2” in front of the real part when converting to complex

Issue 2 - How do you select the best contour geometry for a given complex integral?

You chose a semicircle although there are many other shapes

[kayleeweatherspoon]

I *might* actually pass my qualifying exam because of your videos--thank you so much!

[George-ijgm]

A nice video and logic, but I have some difficulties to recognize your handwrinting of x, u, and n, quite often thery all look the same.

[jperez]

this is how to show a solution, you showed every step and why. it infuriates me to follow a proof and they skip several steps

[shashwat]

Very nice video, helped me a lot. Thank you so much.

[eneXeon]

Nice video! But please do not write = in the two equations from 17:02 on. They are not equal to terms on the line before!

[snipergranola]

Plz upload more videos and concept u just saving our time, question and theorem at the same time, killed 2 birds with one stone

[abcdef]

at 5:53, it is still not logical enough. the limits of integral from -infinity to - epsilon where the changed angle pi which must be implimented, compared to the angle 0 from +eps to + inf.

[simrannahar]

this was such a helpful vide, really really benefitted from it so thank you so much!

[elhabaymohcine]

for tht integr for £ we can use residu simplary -ipi

[lucasciacovelli]

You could add that Theo absolut value of the integral is the Same Than the integral since its zero, also pi and zero are the values that you put into the antiderivative of the integrand so there are no Problems with that sin