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.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Complex analysis and vector fields

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