Complex Analysis: Integral of x/sinh(x)

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  • เผยแพร่เมื่อ 13 ม.ค. 2025

ความคิดเห็น • 18

  • @angelmendez-rivera351
    @angelmendez-rivera351 2 ปีที่แล้ว +11

    Technically, the integral of 1/sinh does not diverge, precisely because in your parametrization, you chose a symmetric interval with a single parameter ε, and the integrand is odd, so the integral is always 0, for all ε and R. So in the limit as ε -> 0, the integral is 0.
    It would have been a problem if the intervals were not symmetric, but then the contour integral over γ would not have been doable. So actually, everything is okay here, and there is no divergence.

  • @fordtimelord8673
    @fordtimelord8673 3 ปีที่แล้ว +5

    Superior teaching, with this, you made something I was unsure about into child’s play (almost). The added bonus at the end is sincerely appreciated.

  • @user-wu8yq1rb9t
    @user-wu8yq1rb9t 2 ปีที่แล้ว +3

    🤓🤓
    Honestly, I didn't expect that! These integral is super cool and this video is actually so great (definitely I'll back and watch it again).
    I learned and enjoyed (especially when you use two methods).
    Thank you so much dear *QN³* ❤️

  • @arielfuxman8868
    @arielfuxman8868 3 ปีที่แล้ว +2

    Your channel is the best.

  • @OvsankaPoutram
    @OvsankaPoutram 2 ปีที่แล้ว +1

    I must be too lucky to have discovered your channel

  • @user-wu8yq1rb9t
    @user-wu8yq1rb9t 2 ปีที่แล้ว

    🤓 I love this video.
    Thank you dear *QN³* for bringing it to us 💗

  • @amath7874
    @amath7874 3 ปีที่แล้ว +1

    Thank you Sir.
    Happy New Year

  • @Walczyk
    @Walczyk 2 ปีที่แล้ว

    I took x to lnx and took partial fractions and simplified to get 2lnx*(1/(x-1) + 1/(x+1)) and the bounds now go from 0 to infinity. I tried integration by parts and it started to look really circular and also divergent, edit: corrected some mistakes.
    Interestingly from zero to 1 the first integral ln(x)/(x-1) equals pi^2/6 exactly! And both integrals combined equal pi^2/4 ! Wow! After everything including the factor of 2 from Sinh you wind up with pi^2 again, and the combined integral from 0 to 1 is equal to the integral from 1 to infinity, and the indefinite integral is in terms of two dilogarithms and the product of two functions

  • @SumitYadav-mx8bp
    @SumitYadav-mx8bp 2 หลายเดือนก่อน

    In the calculation,there was a pole at the origin too, why you didn't avoid it by making a semicircular detour around it, like the one you have made at around z = +iπ ?

    • @qncubed3
      @qncubed3  2 หลายเดือนก่อน

      It is a removable singularity

  • @wupperinteg9177
    @wupperinteg9177 3 ปีที่แล้ว +4

    Dear QN^3, I wish you a happy new year!
    Could you please do the same integral over exp(i*a*x)*cosh(x)/sinh(x).

  • @daniellindner355
    @daniellindner355 2 ปีที่แล้ว

    Incredible !!! Can you recommend any good books for self study in complex analysis?

  • @complexrealmaster5511
    @complexrealmaster5511 3 ปีที่แล้ว

    Happy New Year

  • @user-wu8yq1rb9t
    @user-wu8yq1rb9t 3 ปีที่แล้ว

    Hello Dear *QN^3*
    Happy New Year

  • @aneeshsrinivas9088
    @aneeshsrinivas9088 2 ปีที่แล้ว

    U should have made the height of this rectangle 3π/2