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Math Help - anlytic function

  1. #1
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    anlytic function

    How can I find all analytic functions f=u+iv with u(x,y)=(x^2)+(y^2)

    Thanks for the help. I appreciate it.
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  2. #2
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    Re: anlytic function

    a) do you know what it means for a function to be analytic?
    b) do you know what the Cauchy–Riemann equations are?

    if not read up on those two and give it a shot.
    Thanks from topsquark
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  3. #3
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    Re: anlytic function

    i tackled with the problem which is u(x,y)=(x^2)-(y^2)

    but i need help for that one u(x,y)=(x^2)+(y^2)

    i know your questions. thanks for your help.
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  4. #4
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    Re: anlytic function

    to the best of my knowledge
    if u(x,y)=(x^2)-(y^2) then

    f=(x^2)-(y^2)+ i*2xy
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  5. #5
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    Re: anlytic function

    Quote Originally Posted by mami View Post
    i taclked the problem which is u(x,y)=(x^2)-(y^2)

    but i need help for that one u(x,y)=(x^2)+(y^2)

    i know your questions. thanks for your help.
    ok, so you know that du/dx = dv/dy and du/dy = -dv/dx

    you have u so compute it's partials

    du/dx = 2x
    du/dy = 2y

    so dv/dy = 2x and thus v = 2xy + f(y)

    dv/dx = -du/dy = -2y so v = -2xy + g(x)

    2xy + f(y) = -2xy + g(x)

    4xy = g(x) - f(y), and you can see that this has no solution
    Last edited by romsek; November 29th 2013 at 01:42 PM.
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  6. #6
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    Re: anlytic function

    thanks again. but I have a problem again. could you please check it. is it relevant our question? my mind confused...
    anlytic function-question.jpganlytic function-solution.jpg
    Last edited by mami; November 29th 2013 at 01:47 PM.
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  7. #7
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    Re: anlytic function

    at (0,0) 4xy = 0 so the C-R equations happen to have a solution there so yes (x^2+y^2) + i*0 happens to satisfy the C-R equations at 0.

    so yeah, you can say (x^2+y^2) + i 0 is analytic at 0 but nowhere else on the complex plane.
    Thanks from mami
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  8. #8
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    Re: anlytic function

    would you mind, may i send e-mail to you if i couldn't tackle any problem?
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  9. #9
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    Re: anlytic function

    Quote Originally Posted by romsek View Post
    at (0,0) 4xy = 0 so the C-R equations happen to have a solution there so yes (x^2+y^2) + i*0 happens to satisfy the C-R equations at 0.

    so yeah, you can say (x^2+y^2) + i 0 is analytic at 0 but nowhere else on the complex plane.
    I should be a bit more careful. You need more than satisfying the C-R equations to declare a function is analytic at a point. The first partials must exist and be continuous at that point. The first partials of (x^2+y^2) + i*0 do exist and are continuous everywhere and thus are at (0,0).
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  10. #10
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    Re: anlytic function

    posting them will be faster as I'm not always around.
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