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Math Help - The Cauchy Riemann equations and harmonic functions

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    The Cauchy Riemann equations and harmonic functions

    I have two functions u and v which are defined as harmonic within a region (they both satisfy Laplace's equation)

    If I can show that u and v satisfy the Cauchy Riemann equations along a line in that region, can one show that they must satisfy the C-R everywhere?

    Maybe something like a Taylors expansion of u and v along that line?
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    MHF Contributor Bruno J.'s Avatar
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    No! It is possible for a function of a complex variable to satisfy the C-R equations on a line but nowhere else. Such a function is then nowhere analytic, since it does not satisfy the C-R equations on any nonempty open set.
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    Quote Originally Posted by Bruno J. View Post
    No! It is possible for a function of a complex variable to satisfy the C-R equations on a line but nowhere else. Such a function is then nowhere analytic, since it does not satisfy the C-R equations on any nonempty open set.
    That does not prove or disprove the original statement.
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    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by xxp9 View Post
    That does not prove or disprove the original statement.
    There is no statement in the original post, unless I do not understand what a statement is.

    The post was a question, which I answered.
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    but you didn't use the fact that both u and v are harmonic at all. Harmonic functions have some sort of rigidity like an analytic function.
    So the actual question is, given three equations of v:
    \Delta v = 0, \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
    Where u is a known harmonic function.

    The first equation holds for the whole region, while the last two only hold on a line. Do the 3 equations determine v uniquely( up to a constant)?

    Your argument doesn't prove or disprove this one.

    I guess the answer is yes but I haven't got a proof yet.
    Last edited by xxp9; March 8th 2011 at 09:17 PM.
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    deleted.
    Last edited by xxp9; March 8th 2011 at 09:15 PM.
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    Without loss of generality suppose the line is (part of) the x-axis.
    With the two partial derivatives v_x and v_y known along the x-axis for a harmonic function v, we're expecting to determine v up to a constant.
    Consider the function f=v_x, as the partial derivative of a harmonic function, f itself is harmonic. And since we know f along the x-axis, its partial derivative f_x is known along the x-axis, and since f_y=v_{xy}=v_{yx}, and v_y is known along the x-axis, f_y is also known along the x-axis.
    Similarly, let g=v_y being harmonic, g_x is known and g_y=v_{yy}=-v_{xx}=-f_x is also known, along the x-axis.
    Thus we got two more harmonic functions f and g, with both of their partial derivatives known along the x-axis.
    Proceeding with the same steps, we get all orders of all the partial derivatives of v known along the x-axis. Since v is harmonic v is real analytic, that is, v equals to its Taylor series expansion in a neighborhood of a chosen point.
    Then v is uniquely dermined along a neighborhood of the x-axis, if we choose its value at a point. Use the procedure of analytic continuation we can expand the region without crossing any singular points.
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