Results 1 to 6 of 6

Thread: Find the residue

  1. #1
    Junior Member
    Joined
    Mar 2011
    Posts
    45

    Find the residue

    How to find the residue of
    $\displaystyle \frac{sinh(z)}{cosh(z)-1}$
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32
    Where to we have to find the residue?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,163
    Thanks
    46
    Quote Originally Posted by hurz View Post
    How to find the residue of $\displaystyle \frac{sinh(z)}{cosh(z)-1}$

    If you mean the residue at $\displaystyle z=0$ then, $\displaystyle \dfrac{\sinh z}{\cosh z -1}=\dfrac{z+z^3/3!+\ldots}{ z^2/2!+z^4/4!+\ldots}=\dfrac{2}{z}+\ldots$ . So, the residue is $\displaystyle 2$ .
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member Random Variable's Avatar
    Joined
    May 2009
    Posts
    959
    Thanks
    3

    Re: Find the residue

    If you don't want to use long division, you could use L'Hospital's rule.

    $\displaystyle \lim_{z \to 0} \ (z-0) \ \frac{\sinh z}{\cosh z - 1} = \lim_{z \to 0} \frac{\sinh z + z \cosh z}{\sinh z} $

    $\displaystyle = \lim_{z \to 0} \frac{\cosh z + \cosh z + z \sinh{z}}{\cosh z} = \frac{1+1+0}{1} = 2 $
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Mar 2011
    Posts
    45

    Re: Find the residue

    Quote Originally Posted by FernandoRevilla View Post
    If you mean the residue at $\displaystyle z=0$ then, $\displaystyle \dfrac{\sinh z}{\cosh z -1}=\dfrac{z+z^3/3!+\ldots}{ z^2/2!+z^4/4!+\ldots}=\dfrac{2}{z}+\ldots$ . So, the residue is $\displaystyle 2$ .
    And about the other roots:
    $\displaystyle i(2n\pi + \pi)$ , with n integer

    Regards
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,163
    Thanks
    46

    Re: Find the residue

    Quote Originally Posted by hurz View Post
    And about the other roots: $\displaystyle i(2n\pi + \pi)$ , with n integer
    The roots of $\displaystyle \cosh z -1=0$ are $\displaystyle z_n=2n\pi i$ with $\displaystyle n\in\mathbb{Z}$ . If $\displaystyle n\neq 0$ is better to use the "standard" method.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. residue
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 23rd 2011, 12:17 PM
  2. residue
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Apr 27th 2011, 04:43 AM
  3. find the least positive residue
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: Mar 21st 2010, 11:23 AM
  4. [SOLVED] Find a residue (checking my result and clearing doubts)
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: Oct 19th 2009, 07:12 PM
  5. find the least positive residue
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: May 6th 2008, 06:07 PM

Search Tags


/mathhelpforum @mathhelpforum