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Math Help - Cauchy-Riemman equations

  1. #1
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    Cauchy-Riemman equations

    Ok. I am used to see written everywhere the following Cauchy-Rieamman equations for complex functions:

    and


    Now, today I saw this other notation
    and I'd like to understand how it maps to the first. It must be pretty simple, I guess?

    Thanks
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  2. #2
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    Quote Originally Posted by devouredelysium View Post
    Ok. I am used to see written everywhere the following Cauchy-Rieamman equations for complex functions:

    and


    Now, today I saw this other notation
    and I'd like to understand how it maps to the first. It must be pretty simple, I guess?

    Thanks
    f = u + iv.
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  3. #3
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    Quote Originally Posted by devouredelysium View Post
    Ok. I am used to see written everywhere the following Cauchy-Rieamman equations for complex functions:

    and


    Now, today I saw this other notation
    and I'd like to understand how it maps to the first. It must be pretty simple, I guess?

    Thanks

    Yes it is, but only after one understands what's going on here: you have a complex variable function f(z), but we can put z=x+iy=(x,y), where i=\sqrt{-1} and (x,y) is the representation of a complex number in the complex plane.
    Thus we can write f(z)=f(x,y)=u(x,y)+iv(x,y) , with the usual division of the function in its real and imaginary functions u , v

    Now, finally, applying the rule for partial derivatives of multivariable functions we get:

    i\frac{\partial f}{\partial x}=i\left(\frac{\partial u}{\partial x}+i\,\frac{\partial v}{\partial x}\right)= \left(\frac{\partial u}{\partial y}+i\,\frac{\partial v}{\partial y}\right)=\frac{\partial f}{\partial y}

    Now just compare real and imaginary parts in both sides.

    Tonio
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  4. #4
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    Hmm was almoooost there. Now I get it! Thanks!
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