Cauchy-Riemann --> Laplace equations

Posted on 2007-08-05
Last Modified: 2008-04-03
If a function satisfies the Cauchy-Riemann equations, then it also satisfies the Laplace equation.

(1) Proof/derivation?

(2) Is the reverse also true? (I'm guessing not..)
Question by:InteractiveMind
    LVL 142

    Expert Comment

    by:Guy Hengel [angelIII / a3]
    This looks like homework... sorry, you know the rules, InteractiveMind...
    please come up with some of your thoughts!
    LVL 25

    Author Comment

    haha, I can see why you might think that, but I can assure you it's not.
    I've finished school now, and won't be starting university for another couple of months (plus, it's the summer! ;)).

    I'm just curious.
    LVL 142

    Accepted Solution

    curious: did you already try google or wikipedia?'s_equation
    LVL 11

    Assisted Solution

    LVL 25

    Author Comment

    The Mathworld site provides a simple proof that if a complex function satisfies the Cauchy-Riemann equations, then it also satisfies the Laplace equation.

    And it would appear that the reverse proof results in some additional constants -- so I think the answer to (2) will be a "no".

    However, I do have a question about the derivation provided on the Mathworld page: it seems to make two assumptions:

    i) That u+iv is second differentiable; and
    ii) that D/(DxDy) = D/(DyDx) (D for Delta).

    How to justify these?
    LVL 11

    Assisted Solution

    If you have the time and patience to read the following forum threads, it is an interesting discussion on this subject:

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