So many things to learn from this integral!!

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Опубликовано:

18 сен 2024

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Комментарии : 31

@TheEternalVortex42 11 месяцев назад

I'm kind of confused because f(π) could have been defined as any other value but that didn't seem to matter for computing the a_n coefficients.

@MathFromAlphaToOmega 11 месяцев назад

Rather than doing that, you could leave f(π) undetermined. Then compute the Fourier series, and the value of the Fourier series at x=pi converges to the average of the left- and right-hand limits (that's a general property of Fourier series at jump discontinuities).

@peasant12345 11 месяцев назад

not really, in order to have fourier expansion converge to the function itself pointwisely, the f(pi) must be average of the left limit and right limit at pi

@BikeArea 11 месяцев назад

19:34 Six seconds of silence while writing - very unusual for M. P. 😮

@CM63_France 11 месяцев назад

usually he cuts them during video editing, including the sigh 😉

@carloseliasmartinez6221 11 месяцев назад

Evaluating the Fourier series of the piece-wise expansion at x=pi seems a little bit tricky since you manually defined f(pi)=ln(pi)/2 but it didn't came to be relevant at computing the Fourier coefficients. I guess the choice is quite "natural", yet any I can imagine other choice might have resulted in some delta terms at x=pi.

@MathFromAlphaToOmega 11 месяцев назад

That's really the only choice that makes sense, because if there's a jump discontinuity, the Fourier series will converge to the average of the left- and right-hand limits there.

@GreenMeansGOF 11 месяцев назад

This is exactly what is confusing me. How do we have equality when he just chose what f(π) is?

@lorenzosaudito 11 месяцев назад

@@GreenMeansGOFFourier magic

@ikarienator 11 месяцев назад

That's a precise requirement. A function f(x) that can be expanded into a Fourier series must have the property that at every point, it equals to half of the sum of its left and right limit. Search for Dirichlet-Jordan test.

@alx1984 11 месяцев назад

@@ikarienator What is about x=0? ln(x) -> infinity when x->0 Is this function 'of bounded variation'? (Dirichlet-Jordan test)

@gp-ht7ug 11 месяцев назад

Why has f(x) been defined the way it has been? Not clear to me

@tahirimathscienceonlinetea4273 11 месяцев назад

Hi micheal you deserve a lot for hard working

@nickruffmath 11 месяцев назад

Nice! These integral and sum problems quickly train you to notice 1/(1 - x^2) and swap it with a geometric series, then exchange the order of summation/integration/etc. When you have a 1/(1 + x^2) it can be more annoying because of the (-1)^k that gets added. But it's an interesting tool to recognize that 1/(1 + x^2) is the Laplace transform of sin(x) and replace it with that integral instead. Especially because the only condition is x>0 instead of 0

@BikeArea 11 месяцев назад

When backflip? 😮

@andresomarlegarda616 11 месяцев назад

👌👌

@BikeArea 11 месяцев назад

16:33 actual solving starts

@peasant12345 11 месяцев назад

how about using residue thm?

@RalphLauer 10 месяцев назад

Anybody knows where to find the proof of the first tool?

@mairc9228 11 месяцев назад

what minus sign did you use exacrly? 20:28

@samueldevulder 11 месяцев назад

Laborious, but worth it !

@mcbeaulieu 11 месяцев назад

Desmos gives me a undefined answer. Dunno why 😅

@GreenMeansGOF 11 месяцев назад

(τ/2)·ln(e/2)

@minwithoutintroduction 11 месяцев назад

23:39 رحلة ممتعة

@khayalaliyev3519 11 месяцев назад

This Man Michael Penn is way far beyond other mathematicians

@khayalaliyev3519 11 месяцев назад

@@English_shahriar1 your videos are on beginner level questions for those who watch Michael Penn's videos;if you are from Azərbaycan,we can build up relationship over math

@khayalaliyev3519 11 месяцев назад

@@English_shahriar1 I didn't want to hurt you;you also do a great job!Keep doing 💪

@khayalaliyev3519 11 месяцев назад

@@English_shahriar1 sorry If I hurt you;I felt that in my heart towards you, sorry MAN

@CM63_France 11 месяцев назад

and probably a better climber than them 😉