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Math Help - Complex Analysis - CR equation understanding

  1. #1
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    Complex Analysis - CR equation understanding

    Hi just a few questions on understanding the Cauchy-Riemann equations,

    So for f(x+iy)=(x^2+y)+i(2xy-x)

    I have my CR equations as:

    du/dx=2x , dv/dx=2y-1

    du/dy=1 , dv/dy=2x

    => 2x=2x and 2y-1= -1 , when y=0

    As my first partials are continuous and exist, they satisfy the CR equations when y=0, does that mean the function is differentiable for z= x+i0? implying all real numbers?

    Then does the derivative become

    f'(z)= 2x -1i ?

    There is no answer in the book for this, i just want to make sure i am doing it right

    thanks for your help!!!
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  2. #2
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    Re: Complex Analysis - CR equation understanding

    The derivative exists when \displaystyle y = 0, so that means your function would be \displaystyle f = x^2 - ix, and the derivative is \displaystyle f' = 2x - i.
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  3. #3
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    Re: Complex Analysis - CR equation understanding

    Okay Thanks,

    On a side note, would the right notation for the derivative be, f'(x)=2x-i , \{ x | x  \in \Re\}
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