Results 1 to 4 of 4

Math Help - Proof Cauchy-Goursat

  1. #1
    Member
    Joined
    Oct 2008
    Posts
    156

    Proof Cauchy-Goursat

    Cauchy-Goursat Theorem: If  f(z) is analytic inside and on a simple, closed piecewise smooth curve  C , then  \oint_{C} f(z) \ dz = 0 .

    In proving the "special case" where we use Greens Theorem and the Cauchy Riemann Equations, is assuming that  f is continuously differentiable the same thing as saying that  f has continuous partial derivatives?
    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 manjohn12 View Post
    Cauchy-Goursat Theorem: If  f(z) is analytic inside and on a simple, closed piecewise smooth curve  C , then  \oint_{C} f(z) \ dz = 0 .

    In proving the "special case" where we use Greens Theorem and the Cauchy Riemann Equations, is assuming that  f is continuously differentiable the same thing as saying that  f has continuous partial derivatives?
    If f is continous differenciable (on some open set) it means f is differenciable and f' is continous (on this open set). Write f = g + hi. Remember that f ' = g_x + h_x i. Thus, g_x,h_x are continous on this open set. However, by Cauchy-Riemann equations g_x=h_y and h_x = -g_y so h_y,g_y are continous also. Thus, the partial derivatives must be continous.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2008
    Posts
    156
    Quote Originally Posted by ThePerfectHacker View Post
    If f is continous differenciable (on some open set) it means f is differenciable and f' is continous (on this open set). Write f = g + hi. Remember that f ' = g_x + h_x i. Thus, g_x,h_x are continous on this open set. However, by Cauchy-Riemann equations g_x=h_y and h_x = -g_y so h_y,g_y are continous also. Thus, the partial derivatives must be continous.
    You mean  f_x = g_x +ih_x ?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by manjohn12 View Post
    You mean  f_x = g_x +ih_x ?
    No. The derivative of f is computed to be g_x + ih_x.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Applying Cauchy-Goursat theorem/ doing it the "hard way"
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: July 20th 2010, 12:01 PM
  2. Cauchy-Goursat
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: March 21st 2010, 12:48 AM
  3. cauchy goursat theorm etc
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: February 28th 2010, 07:15 PM
  4. Almost Cauchy-Goursat theorem
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: September 16th 2009, 11:56 AM
  5. Cauchy-Goursat Theorem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 14th 2008, 08:59 AM

Search Tags


/mathhelpforum @mathhelpforum