Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Almost zeta(2)

  1. May 8, 2013 #1

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    I'm noticing wolfram alpha has the amazing ability to analytically solve
    [tex] \sum_{n=1}^{\infty} \frac{1}{n^2 + a^2} [/tex]

    Anyone know how to do this, and if it's also possible to deal with higher order guys (like it also can do 1/(n4+a2), but it's a way more complicated expression to the point where I'm staring at it wondering if it's actually a real number)
     
  2. jcsd
  3. May 8, 2013 #2

    mathman

    User Avatar
    Science Advisor
    Gold Member

    It is a real number. However I have no idea how to calculate it.
     
  4. May 8, 2013 #3
    I'd split it into partial sums and then evaluate.

    Hint:
    ##\frac{1}{n^2+a^2}=\frac{1}{(n+ai)(n-ai)}##, where i is the imaginary unit.

    Higher order expressions will undoubtedly be a lot more complicated. Though, given n and a are real numbers, the sum will also be real, so they won't necessarily be more complex (heh. See what I did there? :tongue:).
     
  5. May 8, 2013 #4

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    If you split it into two fractions of degree one neither of your series converge anymore.

    By "wondering if it's a real number" I meant "gee wolfram alpha has a lot of 4th roots of -1 in that expression" not "I literally don't know if it's a real number"
     
  6. May 9, 2013 #5

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

  7. May 9, 2013 #6

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Oh wow that was easy. I was too busy approaching the problem from the wrong side of the equation. Thanks
     
  8. May 9, 2013 #7

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    It is actually not easy, not to me. I can't find a proof for the series expansion. :D
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook