Let $ f: \mathbb{R}^2 \rightarrow \mathbb{R} $ be some analytic function. I want to say something like, as $ x, y \rightarrow 0 $, the taylor expansion looks like: $$ f(x,y) = a x^2 + b y^2 + \text{(monomials of degree larger than 3)} $$ What's the cleanest way to say this? I was thinking for example, as $ x,y \rightarrow 0 $ $$ f(x,y) = a x^2 + b y^2 + O\left( \sqrt{x^2 + y^2}^3 \right) $$ meaning to say, terms that are bounded by some constant times $ \sqrt{x^2+y^2}^3 $ as $ x,y \rightarrow 0 $ but I am not sure if this standard notation.


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