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Complex Analysis: Cauchy Riemann Equations 2

  1. Sep 28, 2014 #1
    Hi All,

    I was reading through Kreyzeig's Advanced Engineering Mathematics and came across two theorems in Complex Analysis.

    Theorem 1:
    Let f(z) = u(x,y) + iv(x,y) be defined and continuous in some neighborhood of a point z = x+iy and differentiable at z itself.
    Then, at that point, the first-order partial derivatives of u and v exist and satisfy the Cauchy–Riemann equations.

    Theorem 2:
    If two real-valued continuous functions and of two real variables x and y have continuous first partial derivatives that satisfy the Cauchy–Riemann equations in some domain D.
    Then the complex function is analytic in D.


    It seems that the hypothesis of Theorem 1 is similar to the conclusion of Theorem 2. Can these two theorems be modified into one iff statement?

    Thanks.
     
  2. jcsd
  3. Sep 29, 2014 #2
    Theorem 1 states that if a function is complex differentiable at one point, then it has partial derivatives at this point, and these partial derivatives satisfy Cauchy--Riemann equations. BTW, continuity of ##f## in a neighborhood of ##z## is not needed in Theorem 1, ##f## only needs to be defined there and to be complex differentiable at the point ##z##. The proof is very easy, just a trivial computation.

    Theorem 2 says that if partial derivatives exist and continuous in a domain ##D##, then the function is analytic in this domain.

    Of course, one can say that a function is analytic in a domain if and only if the partial derivatives exist, are continuous in the domain and satisfy the Cauchy--Riemann equations there; this is a true theorem. But many authors like to emphasize that complex differentiability at one point implies the Cauchy--Riemann equations at this point.
     
  4. Sep 29, 2014 #3

    WWGD

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    I think the standard example of differentiable but not analytic is ## f(z)=|z| ; |z|:= (x^2+ y^2)^{1/2} ; z=x+iy ##
     
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