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Complex analysis and vector fields

  1. Apr 22, 2015 #1
    I'm going to ask a very general question where I just would want to hear different possible methods that can be thought of in this kind of problem. I am trying to solve a very specific problem with this but I won't talk about that because I don't want someone to give me the answer but ideas for methods in these kinds of problems.

    Consider a vector field ##\vec{F}=(F_x,F_y)## for which holds ##\nabla \times \vec{F} = 0 ##.

    I have to prove that ##F_x=\sum\limits_{m=-\infty}^\infty f(m,x,y)## where ##f## is a function of ##x## and ##y## but also depends on the value of the index. The same for ##F_y(x,y)=\sum\limits_{m=-\infty}^\infty g(m,x,y)##.

    I am supposed to prove that ##F_x=G(x,y)## and ##F_y=H(x,y)## where ##G## and ##H## are pretty alright looking functions.

    I know that someone from my class did a very simple proof of this using complex analysis.

    QUESTION: What kind of different complex analysis techniques can be thought of to use in this case? I know the answers can range really broadly but this is physics undergrad level so it can't be too far fetched.
     
  2. jcsd
  3. Apr 23, 2015 #2
    What is ##F_x##? Is it partial derivative of a scalar function ##F## or simply ##x##-coordinate of the vector field ##\vec F##?
     
  4. Apr 23, 2015 #3
    It's the x-coordinate. I forgot that this notation was often used for the derivative in math.
     
  5. Apr 23, 2015 #4
    Do you also assume that your field is divergence free (i.e. that ##\nabla\cdot \vec F =0##)? Then ##F_x - i F_y## satisfies the Cauchy-Riemann equations and you can use complex analysis. But with your assumptions you can only say that your field has a potential, i.e. it is a gradient of a scalar field.
     
  6. Apr 24, 2015 #5
    Sorry for the late answer. Yes it satisfies zero-divergence, my bad for not mentioning it.
     
  7. Apr 24, 2015 #6
    Then as I said, the function ##\phi= F_x -i F_y## satisfies the Cauchy Riemann equations, i.e. ##\phi## is an analytic function. And analytic functions admit power series representations.

    Where your field is defined? Is it a disc or an annulus?
     
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