Results 1 to 2 of 2
Like Tree1Thanks
  • 1 Post By choovuck

Math Help - convergence exponent

  1. #1
    Newbie
    Joined
    Nov 2012
    From
    italy
    Posts
    9

    convergence exponent

    Let \{z_j\} be the sequence of zeros on an entire function f. We define the convergence exponent of \{z_j\} as
    b=\inf\left\{\lambda>0\ \text{s.t.}\ \sum_{j=1}^{+\infty}\frac{1}{|z_j|^{\lambda}}\ \textrm{converges}\right\}

    Let n(r) be the number of z_j's with |z_j|\leq r. Then the following identity holds:
    b=\limsup_{r\rightarrow +\infty}\frac{\log{\ n(r)}}{\log{r}}

    Do you think i should use Jensen formula to prove this?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Aug 2008
    Posts
    74
    Thanks
    1

    Re: convergence exponent

    don't know if Jensen would help or not, but the proof that I have seen uses formula

    \sum \frac{1}{|z_j|^\lambda} = \int_0^\infty \frac{dn(t)}{t^\lambda}
    Thanks from tenderline
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 18
    Last Post: April 12th 2012, 02:34 PM
  2. Replies: 3
    Last Post: April 27th 2011, 04:53 PM
  3. [SOLVED] Help dividing exponent by exponent
    Posted in the Algebra Forum
    Replies: 2
    Last Post: January 29th 2010, 11:20 AM
  4. Replies: 1
    Last Post: August 19th 2008, 08:41 AM
  5. Replies: 5
    Last Post: July 16th 2008, 11:53 AM

Search Tags


/mathhelpforum @mathhelpforum