• Help the next generation of students in the community for the new syllabus!
    Share your notes and exam papers on our Notes & Resources page
  • We are looking for new moderators to join the Bored of Studies team!
    Click here for details

Higher Level Integration Marathon & Questions (1 Viewer)

omegadot

Active Member
Joined
Oct 15, 2015
Messages
231
Gender
Male
HSC
N/A
Re: Extracurricular Integration Marathon



 
Last edited:

Paradoxica

-insert title here-
Joined
Jun 19, 2014
Messages
2,480
Location
Outside reality
Gender
Male
HSC
2016
Re: Extracurricular Integration Marathon

By inspection, the answer to the first integral is:



Similarly, the answer to the second integral is:



These come from the integral definitions of the Riemann Zeta and Dirichlet Eta functions, respectively, with slight manipulation.
 

Kingom

Member
Joined
Apr 25, 2015
Messages
49
Gender
Male
HSC
2019
Re: Extracurricular Integration Marathon

By inspection, the answer to the first integral is:



Similarly, the answer to the second integral is:



These come from the integral definitions of the Riemann Zeta and Dirichlet Eta functions, respectively, with slight manipulation.
Can you please show working out? I can't really learn from this and what is this "slight manipulation" that you have stated.
 

Paradoxica

-insert title here-
Joined
Jun 19, 2014
Messages
2,480
Location
Outside reality
Gender
Male
HSC
2016
Re: Extracurricular Integration Marathon

Evaluate:



Hence, or otherwise, evaluate:

anyway let's bump this one, been sitting here for a very long time.

the first one is a trivial exercise in contour integration/special functions manipulation

the second is a matter of a straightforward substitution
 

omegadot

Active Member
Joined
Oct 15, 2015
Messages
231
Gender
Male
HSC
N/A
Re: Extracurricular Integration Marathon

Beta Function manipulation is pretty trivial....
Beta function, second derivative with respect to its parameter to get rid of the log squared term, beta function reflection formula is then related to the digamma function (and of course its second derivatives will be needed along the way), then a reflection formula for the digamma function and I guess we are home and hosed but is it not nice to see something different :)
 

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

Top