Integral of e^-x ln(x) from 0 to infinity

In this video, I evaluate the integral from 0 to infinity of e^(-x) ln(x) dx using power series. The answer is quite surprising and unexpected, so check it out! Enjoy!

Note: This video is used in the evaluation of the Vardi integral, which I'll do in a later video

[fengshengqin]

Love it ! I have used Laplace transform and turn it into F(s)=L(lnx)=-1/s (lns+C) and integral=F(1)=minus Euler constant

[sunriselp]

So it just equals to an arbitrary pasta letter? xD I have to say that i am not satisfied with that answer.... :(

[qubix]

You're writing letter γ from right to left... That was so unexpected to me (and that is nice)

[crixboyca]

and you can proof that the derivative of gamma function evaluated at 1 is equal to this integral, i.e. minus the euler mascheroni constant

[chandankar]

I'm glad you mentioned dominated convergence theorem.

[mikety]

Do not forget every tool in your toolbox ... _including_ every possible definition of a term or concept, such as that

e^x = lim_{n->oo} (1 + x/n)^n

just as much as

e^x = sum_{n=0...inf} (x^n)/n!

:)

[jeremyb]

J'adore ce genre d'exercice !
Le résultat est tellement imprévisible, on ne sait jamais ce sur quoi on va tomber : en général il y a pi ou ln(2), là il y a la constante d'Euler-Mascheroni xD

Ça me fait penser à l'intégrale de x*tan(x) de 0 à pi/4 qui fait intervenir pi, ln(2) et la constante de Catalan. Elle a pourtant l'air si simple :

D'ailleurs ça pourrait être intéressant de parler de la constante de Catalan un de ces jours  : )

[monikaherath]

I cannot believe I just discovered your fantastic channel now Dr Peyam. I feel if I found this channel 5 years ago I would have totally dedicated myself to mathematics or gone into research / teaching!

[kalvin]

OMG, by the way I cannot wait to see the evaluation steps of the vardi integral!!! OMG OMG!!!💘✨

[cicik]

okay, first question is that N of bounds of integration and N from e^-x are at first different variables, means they can approach infinity with different speed, why it is correct to make them to one variable?

[ЮрійЯрош-гь]

Shouldn't we have two different limits in the first step?

[yaaryany]

All of this is okay but how does one get the intuition to take the first step?
I mean I get it that it should be intrinsic for a mathematician to have intuition but all of these non elementary integrals are so darn arbitrary that if I just substitute something else other than e^-X, then my whole endeavour would be ruined.
Is there a general way to identify a certain kind of integral and then proceed by a prescribed or atleast preferred method or is it totally mathemagical intuition?

[ozzyfromspace]

Thank you! I tried solving this integral on my own and got tripped up with the telescoping series, because I didn’t realize you could simplify it the way you did. I’m excited to try again 😊

Edit:

If you think of N as an integer, you’re really just showing that a sequence of integrals converges to the Euler-Mascheroni constant, which is interesting from a functional analysis perspective 🙌🏽. I’ve spent like a whole day on this integral, so again, thanks for such a lucid explanation, Dr. Peyam!

[gesucristo]

I’m sorry but how do you show that the difference approaches Euler-Mascheroni constant ?

[alvinlepik]

Must be careful at 2:00, limits of either term must exist separately if we want to write it as a product of limits. In this case all is well. It's surprising the integral is negative :o

[quantumcity]

I'm waiting for this integral...
Thanks Dr. payman ....😘

[romainlin]

I have a question, it's how to prove that the integral converges without calculate it. I have an exercise that I just need the prove the convergence

[shanmugasundaram]

The integral itself is equal to (gamma)' (1) which by definition equal to the Euler's constant(negative). Nevertheless the derivation is interesting.Particularly the cancellation of far away terms in the teloscopic series which depends on N.

[TiahraThankyew]

how can you use the dominated convergence theorum if there isnt a bound for the functions near 0. i beleve monotone convergence might be the theorum you want

[MrRyanroberson]

For as much as I love the constant, I still wonder what it's useful for. Platonic constants don't just pop up and be useless.