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Complex Differentiation

  1. Aug 3, 2009 #1
    1. The problem statement, all variables and given/known data

    Find where;

    f(z) = (z+1)/(z-i)

    is differentiable on the complex plane and find the formulas for f'

    2. Relevant equations

    CR equations;

    if f(z) = u(x,y) + iv(x,y)

    u_x - v_y = 0
    v_x + u_y = 0

    if function is differentiable


    3. The attempt at a solution

    My problem is splitting this into its real and imaginary components. Once I have it in re, im parts I know how to use the CR equations to find whether it's differentiable and then find the derivative.
     
  2. jcsd
  3. Aug 3, 2009 #2

    Cyosis

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    Homework Helper

    To simplify the problem slightly write your fraction as:

    [tex]
    \frac{z+1}{z-i}=\frac{(z-i)+i+1}{z-i}=1+\frac{1+i}{z-i}
    [/tex]

    Now use the definition of z that is [itex]z=x+iy[/itex]. Then multiply top and bottom by the complex conjugate of the denominator.
     
    Last edited: Aug 3, 2009
  4. Aug 3, 2009 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    If the problem were
    [tex]\frac{x+1}{x-1}[/tex]
    with x a real number you could differentiate it using the quotient rule couldn't you?

    Well, the rules for differentiating a function of complex numbers are just the same as for functions of real numbers! Differentiate the above and then replace x by z.
     
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