Results 1 to 5 of 5

Math Help - complex functions

  1. #1
    Newbie
    Joined
    Apr 2006
    Posts
    7

    complex functions

    prove that the equation
    Z^3 + 2Z +4=0
    has no roots in the circle uint.
    thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by unicorn
    prove that the equation
    Z^3 + 2Z +4=0
    has no roots in the circle uint.
    thanks.
    Unit circle would mean that you can express your solutions as z=\cos \theta+i\sin \theta
    Thus,
    (\cos \theta+i\sin \theta)^3+2(\cos \theta+i\sin \theta)+4=0
    Using De'Moiver's Theorem,
    \cos 3\theta+i\sin 3\theta+2\cos \theta+2i\sin \theta+4=0
    Thus, (by definition of complex number equality)
    \left\{ \begin{array}{c}\cos 3\theta+2\cos \theta=-4\\ \sin 3\theta +\sin \theta=0
    But the first equality cannot hold because the minimum cosine is at -1. Thus, even if it were to be at -1 then, its value would be (-1)+2(-1)=-3>-4
    Thus, the first equation the LHS is greater than the RHS-impossible.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2006
    Posts
    7

    Smile

    thanks
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by ThePerfectHacker
    Unit circle would mean that you can express your solutions as z=\cos \theta+i\sin \theta
    Now I had read this to mean in the interior of the unit circle not
    on the unit circle

    If it is the interior of the circle that is of interest then we would
    be looking at an application of Cauchy's integral theorem (presumably),
    and then college/calculus would be the appropriate forum.

    RonL
    Last edited by CaptainBlack; April 25th 2006 at 01:15 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member Rebesques's Avatar
    Joined
    Jul 2005
    From
    At my house.
    Posts
    536
    Thanks
    10
    Apply Rouché's Theorem, for f the given function, and g(z)=-z^3+C, C a real constant you can guess by the hypotheses.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Need some help with complex functions... :(
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: January 23rd 2011, 03:44 AM
  2. Complex Functions
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: September 16th 2009, 07:38 AM
  3. Complex Functions
    Posted in the Calculus Forum
    Replies: 0
    Last Post: March 3rd 2009, 10:41 AM
  4. Complex Functions Help
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 5th 2008, 05:53 PM
  5. Complex Functions
    Posted in the Calculus Forum
    Replies: 1
    Last Post: August 19th 2007, 08:03 AM

Search Tags


/mathhelpforum @mathhelpforum