Complex analysis and vector fields

[Coffee_]

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.

 

[Hawkeye18]

What is $F_x$? Is it partial derivative of a scalar function $F$ or simply $x$-coordinate of the vector field $\vec F$?

 

[Coffee_]

What is $F_x$? Is it partial derivative of a scalar function $F$ or simply $x$-coordinate of the vector field $\vec F$?

It's the x-coordinate. I forgot that this notation was often used for the derivative in math.

 

[Hawkeye18]

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.

 

[Coffee_]

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.

Sorry for the late answer. Yes it satisfies zero-divergence, my bad for not mentioning it.

 

[Hawkeye18]

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?