Results 1 to 3 of 3

Math Help - Complex analysis question...

  1. #1
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5

    Complex analysis question...

    In...

    Cauchy-Riemann Equations -- from Wolfram MathWorld

    ... is written that, given the function of x and y...

     f(x,y) = u(x,y) + i\ v(x,y) (1)

    ... and setting...

    z= x + i\ y \implies dz= dx + i\ dy (2)

    ... is...

    \displaystyle \frac{df}{dz} = \frac{\partial {f}}{\partial {x}}\ \frac{\partial {x}}{\partial {z}} + \frac{\partial {f}}{\partial {y}}\ \frac{\partial {y}}{\partial {x}} = \frac{1}{2}\ (\frac{\partial {f}}{\partial {x}} - i\ \frac{\partial {f}}{\partial {y}}) (3)

    It would for me very interesting to have more detail about the steps to arrive at the result (3). Any help will be greatly appreciated!...

    Kind regards

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

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    As always with partial derivatives, you have to ask what the "other variable(s)" are. When you see the notation \frac{\partial {x}}{\partial {z}}, it is implicitly telling you that x is a function of z and some other (unspecified) variable. The usual convention when dealing with functions of a complex variable is that the "other variable" is the complex conjugate \overline{z}, so that x = \frac12(z+\overline{z}) and y = \frac1{2i}(z-\overline{z}). It's then clear that \frac{\partial {x}}{\partial {z}} = \frac12 and \frac{\partial {y}}{\partial {z}} = \frac1{2i}, and equation (3) follows by using the chain rule. Notice that f is meant to be an analytic function, so it is not dependent on \overline{z}. Thus the derivative \frac{df}{dz} can be written with straight d's rather than as a partial derivative.

    In books on complex function theory, the condition for f to be analytic is sometimes written \frac{\partial f}{\partial\overline{z}} = 0. In fact, the condition \frac{\partial f}{\partial\overline{z}} = 0 is exactly equivalent to the Cauchy–Riemann equations.
    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
    ThankYou very much!... sincerly I never suspected of this 'trap'!... it's a very powerful 'key'!...

    Kind regards

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

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: October 4th 2011, 06:30 AM
  2. Replies: 3
    Last Post: May 25th 2009, 01:38 PM
  3. complex analysis question
    Posted in the Calculus Forum
    Replies: 0
    Last Post: December 9th 2008, 12:00 PM
  4. complex analysis question
    Posted in the Calculus Forum
    Replies: 7
    Last Post: May 11th 2008, 08:13 PM
  5. complex analysis question
    Posted in the Calculus Forum
    Replies: 0
    Last Post: December 2nd 2007, 09:44 AM

Search Tags


/mathhelpforum @mathhelpforum