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Thread: analytic function

  1. #1
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    analytic function

    $\displaystyle f:\mathbb{C}\longrightarrow\mathbb{C} $ is an analytic function and

    $\displaystyle f(x+iy) = u(x,y) + iv(x,y) $

    What geometric relationship exists between the level curves of $\displaystyle u $ and those of $\displaystyle v $?

    Draw a rough sketch of these level curves for the case $\displaystyle f(z) = z^{2 } $to support your answer.

    I haven't really any idea where to start this.
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  2. #2
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    Quote Originally Posted by Tekken View Post
    $\displaystyle f:\mathbb{C}\longrightarrow\mathbb{C} $ is an analytic function and

    $\displaystyle f(x+iy) = u(x,y) + iv(x,y) $

    What geometric relationship exists between the level curves of $\displaystyle u $ and those of $\displaystyle v $?

    Draw a rough sketch of these level curves for the case $\displaystyle f(z) = z^{2 } $to support your answer.

    I haven't really any idea where to start this.
    Hint #1 :
    Spoiler:
    Cauchy-Riemann conditions

    Hint #2 :
    Spoiler:
    Orthogonality of gradients of $\displaystyle u$ and $\displaystyle v$

    Hint #3 :
    Spoiler:
    The gradient is perpendicular to level curves


    Hint #4 : do the example first (it could also be hint #0 if you really don't know where to start).
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  3. #3
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    well is this a start? Im clueless at picturing things in my head as regards graphing and plotting stuff

    $\displaystyle f(z) = z^{2} $ and as $\displaystyle z = x+iy $

    => $\displaystyle f(x+iy) = (x^{2} - y^{2}) + 2xyi $

    let $\displaystyle u = x^{2} - y^{2} $ and $\displaystyle v = 2xy $

    therefore $\displaystyle U_{x} = 2x$ and $\displaystyle V_{y} = 2x $

    so
    $\displaystyle U_{x} = V_{y} $

    and $\displaystyle U_{y} = -2y $ and $\displaystyle V_{x} = 2y $

    so
    $\displaystyle U_{y} = -V_{x}$

    Therefore as the Cauchy Riemann equations are satisfied and the function $\displaystyle f(z) = z^{2} $ is analytic...


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  4. #4
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    Quote Originally Posted by Tekken View Post
    well is this a start? Im clueless at picturing things in my head as regards graphing and plotting stuff

    $\displaystyle f(z) = z^{2} $ and as $\displaystyle z = x+iy $

    => $\displaystyle f(x+iy) = (x^{2} - y^{2}) + 2xyi $

    let $\displaystyle u = x^{2} - y^{2} $ and $\displaystyle v = 2xy $

    therefore $\displaystyle U_{x} = 2x$ and $\displaystyle V_{y} = 2x $

    so
    $\displaystyle U_{x} = V_{y} $

    and $\displaystyle U_{y} = -2y $ and $\displaystyle V_{x} = 2y $

    so
    $\displaystyle U_{y} = -V_{x}$

    Therefore as the Cauchy Riemann equations are satisfied and the function $\displaystyle f(z) = z^{2} $ is analytic...
    What you need to do is plot a few level curves, like $\displaystyle u=-3,-2,-1,1,2,3,\ldots$ and same for $\displaystyle v$. It is easier for $\displaystyle v$: we have $\displaystyle y=\frac{v}{2x}$ (hyperbola). For $\displaystyle u$, if you don't know conics (these are also hyperbola), you can plot the level curves on your calculator or computer (equation $\displaystyle y=\pm\sqrt{x^2-u}$ (either + or -)), together with level curves for v and see what relation there is between both (you'll need an orthonormal basis to see it).
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  5. #5
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    Quote Originally Posted by Laurent View Post
    What you need to do is plot a few level curves, like $\displaystyle u=-3,-2,-1,1,2,3,\ldots$ and same for $\displaystyle v$. It is easier for $\displaystyle v$: we have $\displaystyle y=\frac{v}{2x}$ (hyperbola). For $\displaystyle u$, if you don't know conics (these are also hyperbola), you can plot the level curves on your calculator or computer (equation $\displaystyle y=\pm\sqrt{x^2-u}$ (either + or -)), together with level curves for v and see what relation there is between both (you'll need an orthonormal basis to see it).
    Thanks i really appreciate your help. Unfortunately im not at the stage where i really understand your reply... Would you have any link to a website where i could read up on this sort of question, particularly "level curves"?
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  6. #6
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    Quote Originally Posted by Tekken View Post
    Thanks i really appreciate your help. Unfortunately im not at the stage where i really understand your reply... Would you have any link to a website where i could read up on this sort of question, particularly "level curves"?
    I don't know if this help, but you can look here or rather there.

    For a function $\displaystyle u(x,y)$, the level curves of $\displaystyle u$ are the sets of values $\displaystyle (x,y)$ that give the same value $\displaystyle u(x,y)$: for any $\displaystyle u_0\in\mathbb{R}$, the curve of level $\displaystyle u_0$ is $\displaystyle \{(x,y)\in\mathbb{R}^2|u(x,y)=u_0\}$.
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