Results 1 to 3 of 3

Math Help - Harmonic Conjugates

  1. #1
    Junior Member ginafara's Avatar
    Joined
    May 2007
    Posts
    26

    Harmonic Conjugates

    I have two problems, but I am struggling to put the pieces together.

    First:

    Let v be a harmonic conjugate of u. Show that -u is a harmonic conjugate of v.

    Second:

    Suppose that v is a harmonic conjugate of u and that u is a harmonic conjugate of v. Show that u and v must be constant functions.

    Can some push me in the right direction?

    Thanks in advance
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,386
    Thanks
    1476
    Awards
    1
    Quote Originally Posted by ginafara View Post
    Let v be a harmonic conjugate of u. Show that -u is a harmonic conjugate of v.
    Saying the v be a harmonic conjugate of u means that the function f=u+iv satisfies the Cauchy-Riemann equations.
    Thus we know that u_x  = v_y \,\& \,u_y  =  - v_x .
    If g=v+i(-u) does g satisfy the Cauchy-Riemann equations?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by ginafara View Post
    I have two problems, but I am struggling to put the pieces together.

    First:

    Let v be a harmonic conjugate of u. Show that -u is a harmonic conjugate of v.

    Second:

    Suppose that v is a harmonic conjugate of u and that u is a harmonic conjugate of v. Show that u and v must be constant functions.

    Can some push me in the right direction?

    Thanks in advance
    First:

    So you have f(x, y) = u(x, y) + i v(x, y) is analytic.

    You want to prove that g(x, y) = v(x, y) - i u(x, y) is analytic .......

    Note that g(x, y) = -i [i v(x, y) + u(x, y)] = -i f(x, y).

    Therefore .....

    ------------------------------------------------------------------------------------------------

    Second:

    So you have f(x, y) = u(x, y) + i v(x, y) and g(x, y) = v(x, y) + i u(x, y) are analytic.

    You want to prove that u and v are constants.

    From Cauchy-Riemann relations:

    f analytic:

    \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \, and \, \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.


    g analytic:

    \frac{\partial v}{\partial x} = \frac{\partial u}{\partial y} \, and \, \frac{\partial v}{\partial y} = -\frac{\partial u}{\partial x}.


    Therefore:

    \frac{\partial u}{\partial x} = -\frac{\partial u}{\partial x} \, and \frac{\partial u}{\partial y} = -\frac{\partial u}{\partial y} \, .

    It follows that u = constant.


    \frac{\partial v}{\partial x} = -\frac{\partial v}{\partial x} \, and \frac{\partial v}{\partial y} = -\frac{\partial v}{\partial y} \, .

    It follows that v = constant.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: August 25th 2011, 02:33 PM
  2. No. of conjugates
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: February 1st 2010, 05:10 PM
  3. conjugates
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 8th 2010, 01:56 PM
  4. Conjugates
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 12th 2009, 01:35 AM
  5. conjugates
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: April 27th 2008, 06:02 PM

Search Tags


/mathhelpforum @mathhelpforum