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Thread: 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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    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 $\displaystyle f(z)$, but we can put $\displaystyle z=x+iy=(x,y)$, where $\displaystyle i=\sqrt{-1}$ and $\displaystyle (x,y)$ is the representation of a complex number in the complex plane.
    Thus we can write $\displaystyle f(z)=f(x,y)=u(x,y)+iv(x,y)$ , with the usual division of the function in its real and imaginary functions $\displaystyle u , v$

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

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

    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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