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Math Help - analytic function

  1. #1
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    analytic function

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

     f(x+iy) = u(x,y) + iv(x,y)

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

    Draw a rough sketch of these level curves for the case  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
     f:\mathbb{C}\longrightarrow\mathbb{C} is an analytic function and

     f(x+iy) = u(x,y) + iv(x,y)

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

    Draw a rough sketch of these level curves for the case  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 u and 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

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

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

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

    therefore  U_{x} = 2x and  V_{y} = 2x

    so
    U_{x} = V_{y}

    and  U_{y} = -2y and  V_{x} = 2y

    so
    U_{y} = -V_{x}

    Therefore as the Cauchy Riemann equations are satisfied and the function  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

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

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

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

    therefore  U_{x} = 2x and  V_{y} = 2x

    so
    U_{x} = V_{y}

    and  U_{y} = -2y and  V_{x} = 2y

    so
    U_{y} = -V_{x}

    Therefore as the Cauchy Riemann equations are satisfied and the function  f(z) = z^{2} is analytic...
    What you need to do is plot a few level curves, like u=-3,-2,-1,1,2,3,\ldots and same for v. It is easier for v: we have y=\frac{v}{2x} (hyperbola). For u, if you don't know conics (these are also hyperbola), you can plot the level curves on your calculator or computer (equation 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 u=-3,-2,-1,1,2,3,\ldots and same for v. It is easier for v: we have y=\frac{v}{2x} (hyperbola). For u, if you don't know conics (these are also hyperbola), you can plot the level curves on your calculator or computer (equation 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 u(x,y), the level curves of u are the sets of values (x,y) that give the same value u(x,y): for any u_0\in\mathbb{R}, the curve of level u_0 is \{(x,y)\in\mathbb{R}^2|u(x,y)=u_0\}.
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