Euler's brilliant solution to the Basel problem vs Cauchy's cool residue theorem approach

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Doesn't get much better than Euler vs Cauchy.
Whose side are you on?

Infinite product expansion of sin(x):

Infinite series and the residue theorem:

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Maths 505

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maths_

Suiiii.He did it.Great presentation plus u found a way to combine the basel problem with complex analysis. At this point i may as well say thank you for ur service man ur a legend😂

ΙΗΣΟΥΣΧριστος-θγ

Great. Nice to see how the the residue theorem can be applied to Basel's problem.

Galileosays

the fact that f(z)cot(pi z) has a residue of f(n) at integers where f has no pole is a genius insight that paves the way for so many summation problems (along with the fact that the sum of the residues is 0 if f is O(1/z^2) which this vid seems to leave out)
i think cauchy wins, his method is more rigorous and generalizable, the class of series that can be evaluated with hadamard product is much smaller than those that can be evaluated with cotangent summation

realcirno

Great presentation, thanks! On the Basil problem, there is a different more generalized approach due to Euler in the “spectacular sums” section of the book “The Calculus Gallery”. Definitely worth checking out.

anthony

Very nice proof. But when you substitute by k = 1 at the first method, you forget to raise the power of (-1) to 1. Thank you very much for your amazing effort.

MrWael

3:22 why not multiply the pix^3 term with the x^2 terms??? u are ignoring the x^5 terms?

JirivandenAssem

all videos are released at midnight in India

aravindakannank.s.

Hello, is it possible to integrate x/(e^x-1) from x=0 to x=infty WITHOUT using series expansion? Context: I am collecting the ways to compute sum(1/n^2). After Laplace transform and MCT, I encounter this integral, but it would make me a clown if I use series expansion since it just goes all the way back sum(1/n^2).

Super-gtlk