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Math Help - Cauchy-Reimann Equations

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    Cauchy-Reimann Equations

    If you were asked this question, how would you respond?

    What is the relationship between the Cauchy-Reimann Equations and analytic functions?
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    Quote Originally Posted by jzellt View Post
    If you were asked this question, how would you respond?

    What is the relationship between the Cauchy-Reimann Equations and analytic functions?
    Is this a trick question?
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    Not sure if you're being sarcastic, but this is not a trick question.

    There is going to be an essay portion on my exam, and this question is one of the possibilities...

    Any thoughts?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by jzellt View Post
    Not sure if you're being sarcastic, but this is not a trick question.

    There is going to be an essay portion on my exam, and this question is one of the possibilities...

    Any thoughts?
    I mean, they are basically in love and married ideologically. Are you asking to show how?
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    Quote Originally Posted by jzellt View Post
    If you were asked this question, how would you respond?

    What is the relationship between the Cauchy-Reimann Equations and analytic functions?
    First note, that the man's name was Riemann, not Reimann.
    I would say that, roughly speaking, the Cauchy-Riemann equations are the necessary and sufficient condition for a continously differentiable function \mathbb{R}^2\in (x,y)\mapsto (u(x,y),v(x,y))\in \mathbb{R}^2 to correspond via the definition f: \mathbb{C}\ni x+\mathrm{i}y\mapsto u(x,y)+\mathrm{i}v(x,y)\in \mathbb{C} to an analytic function.

    So, basically, the Cauchy-Riemann equations tell us that we can identify a subset of the continously differentiable functions \mathbb{R}^2\to\mathbb{R}^2, namely the subset of those that additionally satisfy the Cauchy-Riemann equations, with the analytic functions \mathbb{C}\to\mathbb{C}
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    Quote Originally Posted by jzellt View Post
    If you were asked this question, how would you respond?

    What is the relationship between the Cauchy-Reimann Equations and analytic functions?
    Regarding the "relationship" question, certainly there's a well-established relation of consequence between the two.

    A bit devoid of details: Function f analytic in an open set S only if C-R equations hold at every point of S.

    Now can we "strengthen" this relation, i.e., what can be said about the converse relation?
    I think the closest we can get to the converse is by adding an additional hypothesis.
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    Quote Originally Posted by jzellt View Post
    If you were asked this question, how would you respond?

    What is the relationship between the Cauchy-Reimann Equations and analytic functions?
    I would quote the theory that you will find in your textbook. Alternatively (if you have no textbook or class notes), I would quote from a reputable reference found using Google: eg. Cauchy?Riemann equations - Wikipedia, the free encyclopedia. Note: Holomorphic Function -- from Wolfram MathWorld
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  8. #8
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    Quote Originally Posted by PiperAlpha167 View Post
    Regarding the "relationship" question, certainly there's a well-established relation of consequence between the two.

    A bit devoid of details: Function f analytic in an open set S only if C-R equations hold at every point of S.

    Now can we "strengthen" this relation, i.e., what can be said about the converse relation?
    I think the closest we can get to the converse is by adding an additional hypothesis.
    I imagine it would be a good idea for me to state explicitly what I alluded to -- concerning the additional hypothesis -- in my first post.
    (Perhaps it's obvious; but if not, here it is.)
    The additional hypothesis is continuity of the first partial derivatives of u and v.
    So then, the closest we can get to a converse of the implication given in my first post is the following:

    Function f is analytic in an open set S if the first partial derivatives of f are continuous and satisfy the C-R equations at every point of S.
    (Clearly, the "addition" is mediated through conjunction.)

    As far as I can see, that's as close as you can get to the sought after relationship mentioned in your OP.
    It's not quite a logical equivalence relation, but it's fairly close.
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