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Derivative of a complex function in terms of real and imaginary parts.

  1. Jan 4, 2012 #1
    Hi, I wonder if anyone knows when (maybe always?) it is true that, where

    [itex]z=x+iy \text{ and } f : \mathbb{C} \to \mathbb{C} \text{ is expressed as } f=u+iv, \text{ that }
    f'(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}?
    [/itex]
    I'm pretty sure that this is true for [itex]f=exp.[/itex]

    I should be able to find this but searching google for mathematics is a nightmare.
     
  2. jcsd
  3. Jan 4, 2012 #2
    Hi,
    if [itex]f(z)[/itex] is analytical function, you can take derivative in any direction on complex plane of [itex]z[/itex], e.g. take it along real axis [itex]dz = dx[/itex].
     
  4. Jan 4, 2012 #3

    micromass

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    This is indeed true. But remember that a complex differentiable function must satisfy Cauchy-Riemann equations!!

    I suggest looking at Theorem I.5.3 of "Complex Analysis" by Freitag and Busam. A free preview is available at google books.
     
  5. Jan 5, 2012 #4
    Thanks for your help. I should have specified that f be analytic.


    Say I wanted to take the derivative in another direction, not along the real or imaginary axis, how would I do that?
     
    Last edited: Jan 5, 2012
  6. Jan 5, 2012 #5
    Thanks for your help. That looks from the preview like a nice book. I'll see if I can find it in the college library.
     
  7. Jan 5, 2012 #6

    micromass

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    This isn't really a question about real analysis, but rather about multivariable limits.

    Say we have the unit vector [itex](a,b)[/itex]. To find the derivative in that direction, we can do this by

    [tex]a\frac{\partial f}{\partial x}+b\frac{\partial f}{\partial y}[/tex]

    This can be done if the function is differentiable at the point.
     
  8. Jan 5, 2012 #7
    Well, you'll get the same answer beacause f(z) is analytic, it follows from definition. You can use Cauchy-Riemann equations to get different forms of df/dz.
     
  9. Jan 5, 2012 #8

    lavinia

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    A complex differentiable function may be thought of as a real differentiable function whose Jacobian is multiplication by a complex number. As a linear transformation of R^2 a complex number is a rotation followed by a dilation. This gives you the relations between the partial derivatives of f.
     
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