Results 1 to 4 of 4

Math Help - Problem with pole

  1. #1
    Newbie
    Joined
    Apr 2009
    Posts
    15

    Problem with pole

    Consider the function: {cot(z)*coth(z)} /z^3

    does it have a pole at z=3 ? If so, can you prove it ?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,963
    Thanks
    1631
    Quote Originally Posted by Mohit View Post
    Consider the function: {cot(z)*coth(z)} /z^3

    does it have a pole at z=3 ? If so, can you prove it ?
    Well, I suppose one could used the definition of "pole"!

    z_0 is a "pole of order n" for f(z) if and only if the Laurent series for f(z) around z_0 has [tex]z^{-n} with non-zero coefficient but no power lower than -n.

    But the crucial point here is that cot(z) and coth(z) are analytic at z= 3 while 1/z^3 is a rational function with non-zero denominator at z= 3 and so also analytic there. All three functions have Taylor's series (Laurent series with no negative powers) about z= 3 and so there product has a Taylor's series there.

    A more interesting problem would be to show that f(z)= cot(z)coth(z)/z^3 has a pole of order 3 at z= 0.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    Quote Originally Posted by HallsofIvy View Post
    ... a more interesting problem would be to show that f(z)= cot(z)coth(z)/z^3 has a pole of order 3 at z= 0...
    ... are You sure? ...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,963
    Thanks
    1631
    Assuming, of course, that neither tan(x) nor tanh(x) is 0 at x= 0!!!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Pole Balancing
    Posted in the Advanced Applied Math Forum
    Replies: 2
    Last Post: May 7th 2011, 06:01 PM
  2. Height of pole
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: August 24th 2010, 07:22 AM
  3. pole
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: April 19th 2010, 02:28 AM
  4. Pole
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: March 31st 2010, 12:45 PM
  5. pole
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 16th 2009, 11:19 AM

Search Tags


/mathhelpforum @mathhelpforum