SymPy: Can I safely differentiate atan2()?

Question

I would like to obtain a symbolic expression which is the derivative of atan2(y,x), where y and x are some expressions with a variable z. Can I safely assume that diff(atan2(y,x),z) gives me what I want?

In math.stackexchange.com there is a proof that atan2 is continuously differentialable in (-pi,pi), but is it in SymPy?

Reagent

To start with, you can't call sympy.diff on an expression of two variables without telling it what you want to differentiate. Also, continuously differentiable for what in (-pi, pi)? Surely not x or y. Maybe it's continuously differentiable cutting atan on that range (which I think means the variable you're differentiating on is nonnegative and the other is nonzero?). – Lorrianelorrie

@Lorrianelorrie atan2(y,x) is two argument function: by y I mean some expression, as in the case of x. You are right, I forgot the differentiation variable in diff! – Reagent

Answer 1

The partial derivatives of atan2(y, x) with respect to x and y are computed by SymPy as

-y/(x**2 + y**2) 
 x/(x**2 + y**2)

and these expressions are continuous as long as x, y do not turn into zero at once. (Assuming real arguments x, y, of course - I don't think anyone puts complex numbers in atan2).

The above formulas are hardcoded here, so we can be very sure that SymPy will return them.

Margoriemargot