# Theorems about properties of two-dimensional even/odd functions?

As the title mentions, I would like to know the properties of two-dimensional even/odd functions. For instance, suppose that the function $f(x,y)$ is even: $$f(-x,-y) = f(x,y)$$ Then what can we say about the properties of $F(x,y) = \int f(x,y) dy$ and $\partial_x F(x,y)$?

• This is not a very strong sense of evenness so there won’t many interesting properties. The ones that do exist can be geometrically intuited: you can related the partials of $f(x,y)$ to those of $f(-x,-y)$, which in some cases will force properties about $F(0,0)$. – Erick Wong Oct 4 '17 at 20:56

It depends on which antiderivative $F(x,y)$ you use. If you take $F(x,y) = \int_0^y f(x,t)\; dt$, then $F$ is odd. But an arbitrary antiderivative $\int f(x,y)\; dy = F(x,y) + c(x)$ for an arbitrary function $c$, and in general this will be neither odd nor even.
• Thanks for your reply. Assuming that we choose the constant of integration such that $F(x,y)$ is odd (which is always possible if I understand you), is $\partial_x F$ then also even again? Do you have any sources (I couldn't find anything via Google)? – Hunter Oct 4 '17 at 21:01